3554:
3326:
2553:
239:
47:
6558:
6601:
3341:. Draw a straight line, and place the point of the compass on one end of the line, and swing an arc from that point to the other point of the line segment. Repeat with the other side of the line. Finally, connect the point where the two arcs intersect with each end of the line segment.
4225:
is a circle (specifically, the incircle). The triangle of largest area of all those inscribed in a given circle is equilateral, and the triangle of smallest area of all those circumscribed around a given circle is also equilateral. It is the only regular polygon aside from the
3364:, place the point of the compass on the circle and draw another circle with the same radius. The two circles will intersect in two points. An equilateral triangle can be constructed by taking the two centers of the circles and either of the points of intersection.
4154:
4831:
1338:
1200:
any one of the statements in the following nine categories is true. Thus these are properties that are unique to equilateral triangles, and knowing that any one of them is true directly implies that we have an equilateral triangle.
1793:
6374:
2542:
3385:
1879:
6151:
5008:
2912:
states that, if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centers of those equilateral triangles themselves form an equilateral triangle.
1947:
4059:
3695:
5457:
1714:
5373:
6553:
4336:
in the interior of an equilateral triangle, the ratio of the sum of its distances from the vertices to the sum of its distances from the sides is greater than or equal to 2, equality holding when
4021:
1407:
6069:
5996:
2124:
702:
3839:
437:
3439:
6198:
1591:
751:
172:
6242:
3736:
5145:
3961:
635:
555:
6594:
2055:
511:
4054:
4685:
304:
5721:
3549:
1508:
4312:
2178:
1644:
1464:
824:
357:
4207:
3791:
859:
588:
1545:
784:
96:
4473:
4444:
4415:
106:
3073:
470:
6481:
6095:
101:
5657:
2235:
connecting some of the centers. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral. In particular:
1245:
6461:
6434:
6407:
6301:
6274:
1239:
1086:
1059:
1032:
272:
6309:
2838:
1984:
6726:
3145:
2890:
2864:
2803:
2777:
2751:
2584:
896:
6665:
5901:
5858:
5835:
5812:
5789:
5766:
5502:
3287:
3264:
3241:
3218:
3195:
3172:
2725:
2702:
2679:
2656:
2633:
2610:
6696:
5921:
5878:
5743:
5622:
5602:
5582:
5562:
5542:
5522:
5479:
5288:
5268:
5248:
5228:
5205:
5185:
5165:
4913:
4893:
4873:
4853:
4680:
4660:
4640:
4620:
4600:
4580:
4560:
4533:
4513:
4493:
4382:
4354:
4334:
4279:
4259:
3910:
3890:
3870:
3758:
3624:
3604:
3507:
3487:
3459:
3362:
3311:
3119:
3093:
3032:
3012:
2992:
2972:
2952:
2434:
2414:
2394:
2374:
2354:
2334:
2314:
1186:
1166:
1146:
1126:
1106:
1002:
979:
956:
936:
916:
381:
1720:
6667:
equilateral triangles. Specifically for star antiprisms, there are prograde and retrograde (crossed) solutions that join mirrored and non-mirrored parallel
2439:
1005:
6566:
6675:, which is the first true member of the infinite family of antiprisms (the tetrahedron, as a digonal antiprism, is sometimes considered the first).
7554:
6100:
3313:
is on the circumcircle then the sum of the two smaller ones equals the longest and the triangle has degenerated into a line, this case is known as
4918:
3632:
5377:
5293:
6486:
2917:
2282:
A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid.
3968:
7628:
1806:
5926:
3796:
864:
In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors, and the medians to each side coincide.
7923:
Riley, Michael W.; Cochran, David J.; Ballard, John L. (December 1982). "An
Investigation of Preferred Shapes for Warning Labels".
1890:
8567:
3700:
2279:
A triangle is equilateral if and only if any three of the smaller triangles have either the same perimeter or the same inradius.
5012:
3915:
2259:
1650:
7113:
7002:
6776:
5661:
2244:
1193:
516:
6632:. All Platonic solids can inscribe tetrahedra, as well as be inscribed inside tetrahedra. Equilateral triangles also form
6628:. In particular, the tetrahedron, which has four equilateral triangles for faces, can be considered the three-dimensional
8002:
7158:
6986:
6980:
3512:
8597:
7907:
7880:
7847:
7699:
7650:
4284:
1351:
2061:
655:
7174:
6812:
6001:
4149:{\displaystyle A={\frac {1}{2}}ab\times {\frac {\sqrt {3}}{2}}={\frac {\sqrt {3}}{4}}ab={\frac {\sqrt {3}}{4}}a^{2}}
394:
3399:
17:
6167:
1551:
714:
136:
9132:
6208:
4357:
707:
Many of these quantities have simple relationships to the altitude ("h") of each vertex from the opposite side:
7790:
7624:
4238:
602:
522:
114:
3037:
1991:
1882:
481:
4536:
4028:
2899:
5627:
277:
31:
4356:
is the centroid. In no other triangle is there a point for which this ratio is as small as 2. This is the
9127:
8720:
8700:
7452:
4213:, and three rational angles as measured in degrees. It is the only acute triangle that is similar to its
3586:
The legs of either right triangle formed by an altitude of the equilateral triangle are half of the base
1470:
8695:
8652:
8627:
7634:
6870:
6574:
2130:
1604:
1414:
791:
309:
88:
4183:
3763:
831:
560:
6578:
4177:
3374:
The proof that the resulting figure is an equilateral triangle is the first proposition in Book I of
3314:
1514:
1189:
3553:
762:
8755:
8025:
3334:
6249:
If a segment splits an equilateral triangle into two regions with equal perimeters and with areas
8680:
7995:
7684:
7426:
6752:
7839:
7833:
6668:
4449:
4420:
4391:
8705:
8590:
4826:{\displaystyle 3\left(p^{4}+q^{4}+t^{4}+a^{4}\right)=\left(p^{2}+q^{2}+t^{2}+a^{2}\right)^{2}.}
4361:
1333:{\displaystyle {\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}={\frac {\sqrt {25Rr-2r^{2}}}{4Rr}}}
7897:
444:
9106:
9046:
8685:
8539:
8532:
8525:
6791:
6590:
6569:
two dimensional space with six triangles meeting at a vertex, whose dual tessellation is the
6466:
6074:
3384:
2909:
592:
The geometric center of the triangle is the center of the circumscribed and inscribed circles
219:
8064:
8042:
8030:
7872:
7865:
8990:
8760:
8690:
8196:
7709:
7591:
7562:
7331:
7012:
6921:
6839:
6439:
6412:
6385:
6279:
6252:
3375:
3098:
2931:
2240:
1212:
1064:
1037:
1010:
475:
245:
7818:
7725:
7668:
7607:
7036:
6962:
6905:
6855:
3626:
of the equilateral triangle. The height of an equilateral triangle can be found using the
2814:
1960:
8:
9096:
9071:
9041:
9036:
8995:
8710:
8551:
8450:
8200:
7966:
6930:
6705:
6641:
6582:
4218:
4169:
4165:
3627:
3290:
3124:
2869:
2843:
2782:
2756:
2730:
2563:
2248:
2197:
875:
596:
384:
211:
7283:(R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
6647:
5883:
5840:
5817:
5794:
5771:
5748:
5484:
3269:
3246:
3223:
3200:
3177:
3154:
2707:
2684:
2661:
2638:
2615:
2592:
1788:{\displaystyle \sin {\frac {A}{2}}\sin {\frac {B}{2}}\sin {\frac {C}{2}}={\frac {1}{8}}}
9101:
8642:
8420:
8370:
8320:
8277:
8247:
8207:
8170:
7988:
7940:
7802:
7795:
7656:
7595:
7579:
7521:
7502:
7494:
7361:
7300:
7188:
7151:
7127:
7090:
7065:
7024:
6950:
6942:
6893:
6843:
6781:
6699:
6681:
6633:
6586:
5906:
5863:
5728:
5607:
5587:
5567:
5547:
5527:
5507:
5464:
5273:
5253:
5233:
5213:
5190:
5170:
5150:
4898:
4878:
4858:
4838:
4665:
4645:
4625:
4605:
4585:
4565:
4545:
4518:
4498:
4478:
4367:
4339:
4319:
4264:
4244:
3895:
3875:
3855:
3743:
3609:
3589:
3492:
3472:
3444:
3347:
3296:
3104:
3078:
3017:
2997:
2977:
2957:
2937:
2419:
2399:
2379:
2359:
2339:
2319:
2299:
1171:
1151:
1131:
1111:
1091:
987:
964:
941:
921:
901:
366:
207:
7546:
7387:
7232:
78:
9081:
8675:
8583:
8559:
7963:
7944:
7903:
7876:
7843:
7806:
7713:
7695:
7660:
7646:
7599:
7207:
7192:
7180:
7170:
7131:
7119:
7109:
7069:
7054:
7028:
7016:
6998:
6954:
6897:
6885:
6847:
6827:
6748:
4222:
2272:
2204:
119:
68:
7506:
7260:
6637:
3325:
2211:
8610:
8563:
8128:
8117:
8106:
8095:
8086:
8077:
8016:
8012:
7932:
7814:
7721:
7690:
7664:
7638:
7603:
7571:
7490:
7486:
7162:
7146:
7101:
7032:
6990:
6958:
6934:
6901:
6851:
6570:
4210:
3559:
2903:
7740:
4364:, which replaces the perpendicular distances to the sides with the distances from
2808:
As these triangles are equilateral, their altitudes can be rotated to be vertical.
9076:
9056:
9051:
9021:
8740:
8715:
8647:
8153:
8138:
7705:
7587:
7008:
6835:
4214:
2224:
223:
64:
57:
7453:"Curious properties of the circumcircle and incircle of an equilateral triangle"
6678:
As a generalization, the equilateral triangle belongs to the infinite family of
9086:
9066:
9031:
9026:
8657:
8637:
8503:
7936:
6613:
6154:
4385:
4173:
2902:
states that, in any triangle, the three points of intersection of the adjacent
2552:
1197:
238:
182:
178:
6919:
Blundon, W. J. (1963). "On
Certain Polynomials Associated with the Triangle".
9121:
9061:
8912:
8805:
8725:
8667:
8520:
8408:
8401:
8394:
8358:
8351:
8344:
8308:
8301:
7550:
7184:
7123:
7020:
6994:
6889:
6831:
6608:
In three dimensions, equilateral triangles form faces of regular and uniform
6379:
6369:{\displaystyle {\frac {7}{9}}\leq {\frac {A_{1}}{A_{2}}}\leq {\frac {9}{7}}.}
3461:
can be derived directly using the
Pythagorean theorem or using trigonometry.
3368:
959:
7717:
7477:
Minda, D.; Phelps, S. (2008). "Triangles, ellipses, and cubic polynomials".
7097:
6813:"An equivalent form of fundamental triangle inequality and its applications"
4164:
An equilateral triangle is the most symmetrical triangle, having 3 lines of
9091:
8961:
8917:
8881:
8871:
8866:
8460:
7810:
7166:
6786:
6759:
6741:
3849:
3338:
3148:
2231:, which implies that the equilateral triangle is the only triangle with no
645:
7766:
7642:
7105:
6736:
Equilateral triangles have frequently appeared in man made constructions:
9000:
8907:
8886:
8876:
8469:
8430:
8380:
8330:
8287:
8257:
8189:
8175:
6740:
The shape occurs in modern architecture such as the cross-section of the
6625:
6617:
2255:
7498:
9005:
8861:
8851:
8735:
8455:
8439:
8389:
8339:
8296:
8266:
8180:
7583:
6946:
6763:
6621:
2232:
363:
Denoting the common length of the sides of the equilateral triangle as
46:
7925:
Human
Factors: The Journal of the Human Factors and Ergonomics Society
4241:, the equilateral triangle has the smallest ratio of the circumradius
2537:{\displaystyle 4\left(p^{2}+q^{2}+r^{2}\right)\geq x^{2}+y^{2}+z^{2}.}
2266:
8980:
8970:
8947:
8937:
8927:
8856:
8765:
8730:
8511:
8425:
8375:
8325:
8282:
8252:
8221:
7971:
6985:. Dolciani Mathematical Expositions. Vol. 36. Washington, D.C.:
6609:
4231:
2925:
7575:
6938:
4539:
that hold with equality if and only if the triangle is equilateral.
8985:
8975:
8932:
8891:
8820:
8810:
8800:
8619:
8485:
8240:
8236:
8163:
5230:
on the inscribed circle of an equilateral triangle, with distances
2228:
2192:
coincide, and are equal, for (and only for) equilateral triangles:
203:
195:
8575:
7694:(1st ed.). New York: Cambridge University Press. p. 85.
7522:"Equilateral triangles and Kiepert perspectors in complex numbers"
6557:
3329:
Construction of equilateral triangle with compass and straightedge
2866:
can be slid up to show that the altitudes sum to that of triangle
8942:
8922:
8835:
8830:
8825:
8815:
8790:
8745:
8606:
8494:
8464:
8231:
8226:
8217:
8158:
6644:
copies of regular polygons are connected by alternating bands of
6629:
6600:
1874:{\displaystyle T={\frac {a^{2}+b^{2}+c^{2}}{4{\sqrt {3}}}}\quad }
756:
7362:"Non-Euclidean versions of some classical triangle inequalities"
7316:
Conway, J. H., and Guy, R. K., "The only rational triangle", in
7215:
Florida
Atlantic University, Department of Mathematical Sciences
7157:. Classroom Resource Materials. Vol. 37. Washington, D.C.:
2560:
Nearest distances from point P to sides of equilateral triangle
8750:
8434:
8384:
8334:
8291:
8261:
8212:
8148:
6640:
in three-dimensional space. For antiprisms, two (non-mirrored)
4227:
2189:
206:
in which all three sides have the same length. In the familiar
3293:
that the sum of any two of them is greater than the third. If
3121:
is an arbitrary point in the plane of an equilateral triangle
2254:
It is also equilateral if its circumcenter coincides with the
788:
The radius of the circle circumscribing the three vertices is
8795:
7961:
6146:{\displaystyle {\frac {1}{q}}+{\frac {1}{t}}={\frac {1}{y}},}
215:
7329:
Leon
Bankoff and Jack Garfunkel, "The heptagonal triangle",
6868:
6604:
A regular tetrahedron is made of four equilateral triangles.
5003:{\displaystyle p^{2}+q^{2}+t^{2}=3\left(R^{2}+L^{2}\right),}
1942:{\displaystyle T={\frac {\sqrt {3}}{4}}(abc)^{\frac {2}{3}}}
8184:
7055:"A new proof of Euler's inradius - circumradius inequality"
6869:
Dospinescu, G.; Lascu, M.; Pohoata, C.; Letiva, M. (2008).
3690:{\displaystyle \left({\frac {a}{2}}\right)^{2}+h^{2}=a^{2}}
3569:
2921:
982:
129:
7279:
Chakerian, G. D. "A Distorted View of
Geometry." Ch. 7 in
5452:{\displaystyle 16\left(p^{4}+q^{4}+t^{4}\right)=11a^{4}.}
3557:
An equilateral triangle with a side of 2 has a height of
1709:{\displaystyle \cos {A}+\cos {B}+\cos {C}={\frac {3}{2}}}
6878:
6820:
5368:{\displaystyle 4\left(p^{2}+q^{2}+t^{2}\right)=5a^{2},}
6548:{\displaystyle z_{1}+\omega z_{2}+\omega ^{2}z_{3}=0.}
6205:
The ratio of its area to the square of its perimeter,
6006:
6004:
4156:
since all sides of an equilateral triangle are equal.
4033:
3344:
An alternative method is to draw a circle with radius
3333:
An equilateral triangle is easily constructed using a
1952:
141:
7832:
Pelkonen, Eeva-Liisa; Albrecht, Donald, eds. (2006).
6708:
6684:
6650:
6489:
6469:
6442:
6415:
6388:
6312:
6282:
6255:
6211:
6170:
6103:
6077:
5929:
5909:
5886:
5866:
5843:
5820:
5797:
5774:
5751:
5731:
5664:
5630:
5610:
5590:
5570:
5550:
5530:
5510:
5487:
5467:
5380:
5296:
5276:
5256:
5236:
5216:
5193:
5173:
5153:
5015:
4921:
4901:
4881:
4861:
4841:
4688:
4668:
4648:
4628:
4608:
4588:
4568:
4548:
4521:
4501:
4481:
4452:
4423:
4394:
4370:
4342:
4322:
4287:
4267:
4247:
4209:. The integer-sided equilateral triangle is the only
4186:
4062:
4031:
3971:
3918:
3898:
3878:
3858:
3799:
3793:
gives the area formula for the equilateral triangle:
3766:
3746:
3703:
3635:
3612:
3592:
3515:
3495:
3475:
3447:
3402:
3350:
3299:
3272:
3249:
3226:
3203:
3180:
3157:
3151:, then there exists a triangle with sides of lengths
3127:
3107:
3081:
3040:
3020:
3000:
2980:
2960:
2940:
2872:
2846:
2817:
2785:
2759:
2733:
2710:
2687:
2664:
2641:
2618:
2595:
2566:
2442:
2422:
2402:
2382:
2362:
2342:
2322:
2302:
2133:
2064:
1994:
1963:
1893:
1809:
1723:
1653:
1607:
1554:
1517:
1473:
1417:
1354:
1248:
1215:
1174:
1154:
1134:
1114:
1094:
1067:
1040:
1013:
990:
967:
944:
924:
904:
878:
834:
794:
765:
717:
658:
605:
563:
525:
484:
447:
397:
369:
312:
280:
248:
139:
7298:
Posamentier, Alfred S.; Salkind, Charles T. (1996).
6747:
Its applications in flags and heraldry includes the
6244:
is larger than that of any non-equilateral triangle.
7922:
6164:The ratio of its area to the area of the incircle,
4016:{\displaystyle A={\frac {1}{2}}ab\sin 60^{\circ }.}
2920:for triangles states that the triangle of greatest
2275:partition the triangle into six smaller triangles.
2267:
Six triangles formed by partitioning by the medians
640:Denoting the radius of the circumscribed circle as
7864:
7794:
7683:
7411:Gardner, Martin, "Elegant Triangles", in the book
7299:
7297:
7150:
7089:
6720:
6690:
6659:
6547:
6475:
6455:
6428:
6401:
6368:
6295:
6268:
6236:
6192:
6145:
6089:
6063:
5990:
5915:
5895:
5872:
5852:
5829:
5806:
5783:
5760:
5737:
5715:
5651:
5616:
5596:
5576:
5556:
5536:
5516:
5496:
5473:
5451:
5367:
5282:
5262:
5242:
5222:
5199:
5179:
5159:
5139:
5002:
4907:
4887:
4867:
4847:
4825:
4674:
4654:
4634:
4614:
4594:
4574:
4554:
4527:
4507:
4487:
4467:
4438:
4409:
4376:
4348:
4328:
4306:
4273:
4253:
4201:
4148:
4048:
4015:
3955:
3904:
3884:
3864:
3833:
3785:
3752:
3730:
3689:
3618:
3598:
3543:
3501:
3481:
3453:
3433:
3356:
3305:
3281:
3258:
3235:
3212:
3189:
3166:
3139:
3113:
3087:
3067:
3026:
3006:
2986:
2966:
2946:
2884:
2858:
2832:
2797:
2771:
2745:
2719:
2696:
2673:
2650:
2627:
2604:
2578:
2536:
2428:
2408:
2388:
2368:
2348:
2328:
2308:
2172:
2118:
2049:
1978:
1941:
1873:
1787:
1708:
1638:
1585:
1539:
1502:
1458:
1401:
1332:
1233:
1180:
1160:
1140:
1120:
1100:
1080:
1053:
1026:
996:
973:
950:
930:
910:
890:
853:
818:
778:
745:
696:
629:
582:
549:
505:
464:
431:
375:
351:
298:
266:
166:
30:"Equilateral" redirects here. For other uses, see
6982:When less is more. Visualizing basic inequalities
6811:Bencze, Mihály; Wu, Hui-Hua; Wu, Shan-He (2008).
3965:Each angle of an equilateral triangle is 60°, so
3367:In both methods a by-product is the formation of
1402:{\displaystyle s=2R+\left(3{\sqrt {3}}-4\right)r}
461:
9119:
7831:
7789:
7427:"Cyclic Averages of Regular Polygonal Distances"
7261:"Inequalities proposed in "Crux Mathematicorum""
7144:
7087:
6561:The equilateral triangle tiling fills the plane.
6483:of 1 the triangle is equilateral if and only if
6064:{\textstyle {\tfrac {t^{3}-q^{3}}{t^{2}-q^{2}}}}
5991:{\displaystyle z={\frac {t^{2}+tq+q^{2}}{t+q}},}
3464:
2119:{\displaystyle r={\frac {r_{a}+r_{b}+r_{c}}{9}}}
697:{\displaystyle A={\frac {3{\sqrt {3}}}{4}}R^{2}}
7545:
7293:
7291:
7289:
5207:and the centroid of the equilateral triangle.
4535:being the vertices). There are numerous other
3834:{\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}.}
2292:A triangle is equilateral if and only if, for
2227:of an equilateral triangle coincides with its
2218:
432:{\displaystyle A={\frac {\sqrt {3}}{4}}a^{2},}
8591:
7996:
7862:
7320:, 1996, Springer-Verlag, pp. 201 and 228–239.
6871:"An elementary proof of Blundon's inequality"
3434:{\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}}
3391:
242:An equilateral triangle. It has equal sides (
222:to each other and are each 60°. It is also a
7863:White, Steven F.; Calderón, Esthela (2008).
7424:
7388:"Another proof of the Erdős–Mordell Theorem"
7359:
7347:100 Great Problems of Elementary Mathematics
7275:
7273:
6978:
6193:{\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}
3852:, the area of a triangle with any two sides
2239:A triangle is equilateral if any two of the
1586:{\displaystyle s={\frac {3{\sqrt {3}}}{2}}R}
755:The height of the center from each side, or
746:{\displaystyle A={\frac {h^{2}}{\sqrt {3}}}}
167:{\displaystyle {\tfrac {\sqrt {3}}{4}}a^{2}}
7619:
7617:
7476:
7286:
7083:
7081:
7079:
6616:are composed of equilateral triangles: the
6237:{\displaystyle {\frac {1}{12{\sqrt {3}}}},}
3469:The area of a triangle is half of one side
8598:
8584:
8003:
7989:
7570:(5). Taylor & Francis, Ltd.: 231–234.
6810:
3731:{\displaystyle h={\frac {\sqrt {3}}{2}}a.}
2954:in an equilateral triangle with distances
7895:
7270:
7088:Andreescu, Titu; Andrica, Dorian (2006).
6979:Alsina, Claudi; Nelsen, Roger B. (2009).
6731:
6597:constructed with equilateral triangles.
5140:{\displaystyle p^{4}+q^{4}+t^{4}=3\left,}
3956:{\displaystyle A={\frac {1}{2}}ab\sin C.}
3320:
2727:, respectively, define smaller triangles
630:{\displaystyle h={\frac {\sqrt {3}}{2}}a}
550:{\displaystyle r={\frac {\sqrt {3}}{6}}a}
460:
7899:Historical Dictionary of the Philippines
7745:Polytopes & their Incidence Matrices
7681:
7614:
7360:Svrtan, Dragutin; Veljan, Darko (2012).
7233:"The vertex-midpoint-centroid triangles"
7076:
7048:
7046:
6974:
6972:
6599:
6556:
3552:
3324:
2551:
2258:, or if its incenter coincides with its
2050:{\displaystyle 9R^{2}=a^{2}+b^{2}+c^{2}}
506:{\displaystyle R={\frac {a}{\sqrt {3}}}}
237:
8568:List of regular polytopes and compounds
7623:
7255:
7253:
7226:
7224:
7052:
6918:
4049:{\displaystyle {\tfrac {\sqrt {3}}{2}}}
233:
14:
9120:
7418:
7344:
2376:to the triangle's sides and distances
2286:
828:The radius of the inscribed circle is
299:{\displaystyle \alpha =\beta =\gamma }
8579:
7962:
7230:
7043:
6969:
6463:, then for either non-real cube root
5716:{\displaystyle q^{2}+qt+t^{2}=a^{2}.}
3843:
7738:
7250:
7221:
6777:Almost-equilateral Heronian triangle
6671:. The Platonic octahedron is also a
5504:of the circumcircle, with distances
2934:states that, for any interior point
867:
8605:
7741:"n-antiprism with winding number d"
7685:"Chapter 2: The Archimedean solids"
7519:
7385:
7205:
7159:Mathematical Association of America
6987:Mathematical Association of America
4234:inside any other regular polygon.
4172:of order 3 about its center, whose
4159:
3544:{\displaystyle A={\frac {1}{2}}ah.}
2547:
1953:Circumradius, inradius, and exradii
1503:{\displaystyle s^{2}=3{\sqrt {3}}T}
24:
7838:. Yale University Press. pp.
7450:
7444:
7323:
7310:
6565:Notably, the equilateral triangle
4453:
4424:
4395:
4217:(with vertices at the feet of the
4189:
3383:
2556:Visual proof of Viviani's theorem
210:, an equilateral triangle is also
25:
9144:
7955:
7835:Eero Saarinen: Shaping the Future
7434:International Journal of Geometry
6200:, is the largest of any triangle.
4307:{\displaystyle {\frac {R}{r}}=2.}
3606:, and the hypotenuse is the side
2173:{\displaystyle r_{a}=r_{b}=r_{c}}
1639:{\displaystyle A=B=C=60^{\circ }}
1459:{\displaystyle s^{2}=3r^{2}+12Rr}
819:{\displaystyle R={\frac {2h}{3}}}
352:{\displaystyle h_{a}=h_{b}=h_{c}}
226:, so it is also referred to as a
7902:. Scarecrow Press. p. 161.
7867:Culture and Customs of Nicaragua
7764:
7302:Challenging Problems in Geometry
5167:is the circumscribed radius and
4202:{\displaystyle \mathrm {D} _{3}}
3786:{\displaystyle {\frac {1}{2}}ah}
2183:
1343:
854:{\displaystyle r={\frac {h}{3}}}
583:{\displaystyle r={\frac {R}{2}}}
104:
99:
94:
45:
7916:
7889:
7856:
7825:
7783:
7758:
7747:. bendwavy.org (Anton Sherwood)
7732:
7675:
7539:
7513:
7470:
7405:
7379:
7353:
7349:. Dover Publ. pp. 379–380.
7338:
6378:If a triangle is placed in the
4221:), and the only triangle whose
3075:independent of the location of
1870:
1540:{\displaystyle s=3{\sqrt {3}}r}
7896:Guillermo, Artemio R. (2012).
7630:Geometries and Transformations
7491:10.1080/00029890.2008.11920581
7199:
7153:Methods for Euclidean Geometry
7138:
6912:
6862:
6804:
6758:It is a shape of a variety of
5187:is the distance between point
4360:; a stronger variant of it is
2906:form an equilateral triangle.
1925:
1912:
779:{\displaystyle {\frac {h}{3}}}
214:; that is, all three internal
13:
1:
7555:"Tilings by Regular Polygons"
7479:American Mathematical Monthly
7425:Meskhishvili, Mamuka (2021).
7208:"Notes on Euclidean Geometry"
7145:Owen, Byer; Felix, Lazebnik;
7092:Complex Numbers from A to...Z
6797:
6160:For an equilateral triangle:
4855:in the plane, with distances
4562:in the plane, with distances
3465:Using the Pythagorean theorem
2924:among all those with a given
2840:is a parallelogram, triangle
2316:in the plane, with distances
1196:respectively, is equilateral
383:, we can determine using the
7767:"Stella Polyhedral Glossary"
7096:(1st ed.). Boston, MA:
3014:from the sides and altitude
2271:For any triangle, the three
652:The area of the triangle is
32:Equilateral (disambiguation)
27:Shape with three equal sides
7:
7871:. Greenwood Press. p.
7682:Cromwell, Peter T. (1997).
7633:(1st ed.). Cambridge:
7335:46 (1), January 1973, 7–19,
6826:(1): 1–6 (Article No. 16).
6770:
4211:triangle with integer sides
2219:Coincident triangle centers
10:
9149:
8557:
7984:
7937:10.1177/001872088202400610
7635:Cambridge University Press
6884:(4): 1-3 (Paper No. 100).
6595:semi-regular tessellations
4468:{\displaystyle \angle CPA}
4439:{\displaystyle \angle BPC}
4410:{\displaystyle \angle APB}
3392:Derivation of area formula
2900:Morley's trisector theorem
599:(height) from any side is
29:
9014:
8960:
8900:
8844:
8783:
8774:
8666:
8618:
7345:Dörrie, Heinrich (1965).
7062:Gazeta Matematica Seria B
4178:dihedral group of order 6
1596:
1148:respectively), and where
644:, we can determine using
177:
128:
113:
87:
77:
63:
53:
44:
39:
7100:. pp. 70, 113–115.
7053:Pohoata, Cosmin (2010).
6995:10.5948/upo9781614442028
6630:analogue of the triangle
4384:to the points where the
4358:Erdős–Mordell inequality
3441:in terms of side length
3335:straightedge and compass
3068:{\displaystyle d+e+f=h,}
2918:isoperimetric inequality
1204:
465:{\displaystyle p=3a\,\!}
306:), and equal altitudes (
7520:Dao, Thanh Oai (2015).
7451:De, Prithwijit (2008).
6753:flag of the Philippines
6476:{\displaystyle \omega }
6090:{\displaystyle t\neq q}
1798:
89:Coxeter–Dynkin diagrams
9133:Constructible polygons
7967:"Equilateral Triangle"
7803:Methuen & Co. LTD.
7801:(1 ed.). London:
7637:. pp. xv, 1–438.
7231:Cerin, Zvonko (2004).
7167:10.5860/choice.48-3331
6732:In culture and society
6722:
6692:
6661:
6605:
6562:
6549:
6477:
6457:
6430:
6403:
6382:with complex vertices
6370:
6297:
6270:
6238:
6194:
6147:
6091:
6065:
5992:
5917:
5897:
5874:
5854:
5831:
5808:
5785:
5762:
5739:
5717:
5653:
5652:{\displaystyle p=q+t,}
5618:
5598:
5578:
5558:
5538:
5518:
5498:
5475:
5453:
5369:
5284:
5264:
5244:
5224:
5201:
5181:
5161:
5141:
5004:
4909:
4889:
4869:
4849:
4827:
4676:
4656:
4636:
4616:
4596:
4576:
4556:
4529:
4509:
4489:
4469:
4440:
4411:
4378:
4350:
4330:
4308:
4275:
4255:
4203:
4150:
4050:
4017:
3957:
3906:
3886:
3866:
3835:
3787:
3760:into the area formula
3754:
3732:
3691:
3620:
3600:
3583:
3545:
3503:
3483:
3455:
3435:
3388:
3358:
3330:
3321:Geometric construction
3315:Van Schooten's theorem
3307:
3283:
3260:
3237:
3214:
3191:
3168:
3141:
3115:
3089:
3069:
3028:
3008:
2988:
2968:
2948:
2896:
2886:
2860:
2834:
2799:
2773:
2747:
2721:
2698:
2675:
2652:
2629:
2606:
2580:
2538:
2430:
2410:
2390:
2370:
2350:
2330:
2310:
2174:
2120:
2051:
1980:
1943:
1875:
1789:
1710:
1640:
1587:
1541:
1504:
1460:
1403:
1334:
1235:
1182:
1162:
1142:
1122:
1102:
1082:
1055:
1028:
998:
975:
952:
932:
912:
892:
855:
820:
780:
747:
698:
631:
584:
551:
507:
466:
433:
377:
360:
353:
300:
268:
168:
7643:10.1017/9781316216477
7460:Mathematical Spectrum
7106:10.1007/0-8176-4449-0
6792:Trilinear coordinates
6723:
6693:
6662:
6603:
6560:
6550:
6478:
6458:
6456:{\displaystyle z_{3}}
6431:
6429:{\displaystyle z_{2}}
6404:
6402:{\displaystyle z_{1}}
6371:
6298:
6296:{\displaystyle A_{2}}
6271:
6269:{\displaystyle A_{1}}
6239:
6195:
6148:
6092:
6066:
5993:
5918:
5898:
5875:
5855:
5832:
5809:
5786:
5763:
5740:
5718:
5654:
5619:
5599:
5579:
5559:
5539:
5519:
5499:
5476:
5454:
5370:
5285:
5265:
5245:
5225:
5202:
5182:
5162:
5142:
5005:
4910:
4890:
4870:
4850:
4828:
4677:
4657:
4637:
4617:
4597:
4577:
4557:
4537:triangle inequalities
4530:
4510:
4490:
4470:
4441:
4412:
4379:
4351:
4331:
4309:
4281:of any triangle, with
4276:
4256:
4204:
4151:
4051:
4018:
3958:
3907:
3887:
3867:
3836:
3788:
3755:
3733:
3692:
3621:
3601:
3556:
3546:
3504:
3484:
3456:
3436:
3387:
3359:
3328:
3308:
3284:
3261:
3238:
3215:
3192:
3169:
3142:
3116:
3090:
3070:
3029:
3009:
2989:
2969:
2949:
2887:
2861:
2835:
2800:
2774:
2748:
2722:
2699:
2676:
2653:
2630:
2607:
2581:
2555:
2539:
2431:
2411:
2391:
2371:
2351:
2331:
2311:
2175:
2121:
2052:
1981:
1944:
1876:
1790:
1711:
1641:
1588:
1542:
1505:
1461:
1404:
1335:
1236:
1234:{\displaystyle a=b=c}
1188:are the radii of the
1183:
1163:
1143:
1123:
1103:
1083:
1081:{\displaystyle r_{c}}
1056:
1054:{\displaystyle r_{b}}
1029:
1027:{\displaystyle r_{a}}
999:
976:
953:
933:
913:
893:
856:
821:
781:
748:
699:
632:
585:
552:
508:
467:
434:
378:
354:
301:
269:
267:{\displaystyle a=b=c}
241:
169:
8831:Nonagon/Enneagon (9)
8761:Tangential trapezoid
7563:Mathematics Magazine
7485:(October): 679–689.
7332:Mathematics Magazine
6989:. pp. 71, 155.
6931:Taylor & Francis
6922:Mathematics Magazine
6706:
6682:
6673:triangular antiprism
6648:
6612:. Three of the five
6487:
6467:
6440:
6413:
6386:
6310:
6280:
6253:
6209:
6168:
6101:
6075:
6002:
5927:
5907:
5884:
5864:
5841:
5818:
5795:
5772:
5749:
5729:
5662:
5628:
5608:
5588:
5568:
5548:
5528:
5508:
5485:
5465:
5378:
5294:
5274:
5254:
5234:
5214:
5191:
5171:
5151:
5013:
4919:
4899:
4879:
4859:
4839:
4686:
4666:
4646:
4626:
4606:
4586:
4566:
4546:
4519:
4499:
4479:
4450:
4421:
4392:
4368:
4340:
4320:
4285:
4265:
4245:
4184:
4060:
4029:
3969:
3916:
3896:
3876:
3856:
3797:
3764:
3744:
3701:
3633:
3610:
3590:
3513:
3493:
3473:
3445:
3400:
3348:
3297:
3270:
3247:
3224:
3201:
3178:
3155:
3125:
3105:
3079:
3038:
3018:
2998:
2978:
2958:
2938:
2870:
2844:
2833:{\displaystyle PGCH}
2815:
2783:
2757:
2731:
2708:
2685:
2662:
2639:
2616:
2593:
2564:
2440:
2420:
2400:
2380:
2360:
2340:
2320:
2300:
2131:
2062:
1992:
1979:{\displaystyle R=2r}
1961:
1891:
1807:
1721:
1651:
1605:
1552:
1515:
1471:
1415:
1352:
1246:
1213:
1172:
1152:
1132:
1112:
1092:
1065:
1038:
1011:
988:
965:
942:
922:
902:
876:
832:
792:
763:
715:
656:
603:
561:
523:
482:
476:circumscribed circle
445:
395:
367:
310:
278:
246:
234:Principal properties
200:equilateral triangle
137:
40:Equilateral triangle
8943:Megagon (1,000,000)
8711:Isosceles trapezoid
8552:pentagonal polytope
8451:Uniform 10-polytope
8011:Fundamental convex
7739:Klitzing, Richard.
7529:Forum Geometricorum
7413:Mathematical Circus
7395:Forum Geometricorum
7386:Lee, Hojoo (2001).
7369:Forum Geometricorum
7318:The Book of Numbers
7240:Forum Geometricorum
7161:. pp. 36, 39.
6721:{\displaystyle n=2}
6636:as well as uniform
5725:Moreover, if point
5290:from the vertices,
4915:from the vertices,
4362:Barrow's inequality
4170:rotational symmetry
4025:The sine of 60° is
3628:Pythagorean theorem
3291:triangle inequality
3140:{\displaystyle ABC}
2885:{\displaystyle ABC}
2859:{\displaystyle PHE}
2798:{\displaystyle PDG}
2772:{\displaystyle PFI}
2746:{\displaystyle PHE}
2579:{\displaystyle ABC}
2287:Points in the plane
2214:have equal lengths.
2207:have equal lengths.
2200:have equal lengths.
898:that has the sides
891:{\displaystyle ABC}
385:Pythagorean theorem
9128:Types of triangles
8913:Icositetragon (24)
8421:Uniform 9-polytope
8371:Uniform 8-polytope
8321:Uniform 7-polytope
8278:Uniform 6-polytope
8248:Uniform 5-polytope
8208:Uniform polychoron
8171:Uniform polyhedron
8019:in dimensions 2–10
7964:Weisstein, Eric W.
7805:pp. 120–121.
7625:Johnson, Norman W.
7281:Mathematical Plums
7206:Yiu, Paul (1998).
6782:Isosceles triangle
6718:
6688:
6660:{\displaystyle 2n}
6657:
6634:uniform antiprisms
6606:
6563:
6545:
6473:
6453:
6426:
6399:
6366:
6293:
6266:
6234:
6190:
6143:
6087:
6061:
6059:
5998:which also equals
5988:
5913:
5896:{\displaystyle PD}
5893:
5870:
5853:{\displaystyle DA}
5850:
5830:{\displaystyle DA}
5827:
5807:{\displaystyle PD}
5804:
5784:{\displaystyle PA}
5781:
5761:{\displaystyle BC}
5758:
5735:
5713:
5649:
5614:
5594:
5574:
5554:
5534:
5514:
5497:{\displaystyle BC}
5494:
5471:
5449:
5365:
5280:
5260:
5240:
5220:
5197:
5177:
5157:
5137:
5000:
4905:
4885:
4865:
4845:
4823:
4672:
4652:
4632:
4622:from the vertices
4612:
4592:
4572:
4552:
4525:
4505:
4485:
4465:
4436:
4407:
4374:
4346:
4326:
4304:
4271:
4251:
4239:Euler's inequality
4199:
4146:
4046:
4044:
4013:
3953:
3902:
3882:
3862:
3844:Using trigonometry
3831:
3783:
3750:
3728:
3687:
3616:
3596:
3584:
3541:
3499:
3479:
3451:
3431:
3389:
3354:
3331:
3303:
3282:{\displaystyle PC}
3279:
3259:{\displaystyle PB}
3256:
3236:{\displaystyle PA}
3233:
3213:{\displaystyle PC}
3210:
3190:{\displaystyle PB}
3187:
3167:{\displaystyle PA}
3164:
3137:
3111:
3085:
3065:
3024:
3004:
2984:
2964:
2944:
2910:Napoleon's theorem
2897:
2882:
2856:
2830:
2795:
2769:
2743:
2720:{\displaystyle CA}
2717:
2697:{\displaystyle BC}
2694:
2674:{\displaystyle AB}
2671:
2651:{\displaystyle HI}
2648:
2628:{\displaystyle FG}
2625:
2605:{\displaystyle DE}
2602:
2576:
2534:
2426:
2406:
2386:
2366:
2346:
2326:
2306:
2170:
2116:
2047:
1976:
1939:
1871:
1785:
1706:
1636:
1583:
1537:
1500:
1456:
1399:
1330:
1231:
1178:
1158:
1138:
1118:
1098:
1078:
1051:
1024:
994:
971:
948:
928:
908:
888:
851:
816:
776:
743:
694:
627:
580:
547:
515:The radius of the
503:
474:The radius of the
462:
429:
373:
361:
349:
296:
264:
208:Euclidean geometry
164:
152:
9115:
9114:
8956:
8955:
8933:Myriagon (10,000)
8918:Triacontagon (30)
8882:Heptadecagon (17)
8872:Pentadecagon (15)
8867:Tetradecagon (14)
8806:Quadrilateral (4)
8676:Antiparallelogram
8573:
8572:
8560:Polytope families
8017:uniform polytopes
7797:Regular Polytopes
7553:(November 1977).
7551:Shepard, Geoffrey
7147:Deirdre, Smeltzer
7115:978-0-8176-4449-9
7004:978-0-88385-342-9
6749:flag of Nicaragua
6691:{\displaystyle n}
6361:
6348:
6321:
6229:
6226:
6188:
6185:
6138:
6125:
6112:
6058:
5983:
5916:{\displaystyle y}
5873:{\displaystyle z}
5738:{\displaystyle D}
5617:{\displaystyle C}
5597:{\displaystyle B}
5577:{\displaystyle A}
5557:{\displaystyle t}
5537:{\displaystyle q}
5517:{\displaystyle p}
5481:on the minor arc
5474:{\displaystyle P}
5283:{\displaystyle t}
5263:{\displaystyle q}
5243:{\displaystyle p}
5223:{\displaystyle P}
5200:{\displaystyle P}
5180:{\displaystyle L}
5160:{\displaystyle R}
4908:{\displaystyle t}
4888:{\displaystyle q}
4868:{\displaystyle p}
4848:{\displaystyle P}
4675:{\displaystyle C}
4655:{\displaystyle B}
4635:{\displaystyle A}
4615:{\displaystyle t}
4595:{\displaystyle q}
4575:{\displaystyle p}
4555:{\displaystyle P}
4528:{\displaystyle C}
4508:{\displaystyle B}
4488:{\displaystyle A}
4475:cross the sides (
4377:{\displaystyle P}
4349:{\displaystyle P}
4329:{\displaystyle P}
4296:
4274:{\displaystyle r}
4254:{\displaystyle R}
4223:Steiner inellipse
4134:
4130:
4113:
4109:
4098:
4094:
4077:
4043:
4039:
3986:
3933:
3905:{\displaystyle C}
3885:{\displaystyle b}
3865:{\displaystyle a}
3816:
3812:
3775:
3753:{\displaystyle h}
3720:
3716:
3649:
3619:{\displaystyle a}
3599:{\displaystyle a}
3530:
3502:{\displaystyle h}
3489:times the height
3482:{\displaystyle a}
3454:{\displaystyle a}
3419:
3415:
3396:The area formula
3357:{\displaystyle r}
3337:, because 3 is a
3306:{\displaystyle P}
3114:{\displaystyle P}
3099:Pompeiu's theorem
3088:{\displaystyle P}
3027:{\displaystyle h}
3007:{\displaystyle f}
2987:{\displaystyle e}
2967:{\displaystyle d}
2947:{\displaystyle P}
2932:Viviani's theorem
2916:A version of the
2436:to its vertices,
2429:{\displaystyle z}
2409:{\displaystyle y}
2389:{\displaystyle x}
2369:{\displaystyle r}
2349:{\displaystyle q}
2329:{\displaystyle p}
2309:{\displaystyle P}
2260:nine-point center
2114:
1936:
1910:
1906:
1868:
1865:
1783:
1770:
1754:
1738:
1704:
1578:
1572:
1532:
1495:
1383:
1328:
1316:
1283:
1270:
1257:
1181:{\displaystyle r}
1161:{\displaystyle R}
1141:{\displaystyle c}
1121:{\displaystyle b}
1101:{\displaystyle a}
997:{\displaystyle T}
974:{\displaystyle s}
951:{\displaystyle c}
931:{\displaystyle b}
911:{\displaystyle a}
868:Characterizations
849:
814:
774:
741:
740:
682:
676:
622:
618:
578:
542:
538:
501:
500:
441:The perimeter is
414:
410:
376:{\displaystyle a}
274:), equal angles (
192:
191:
151:
147:
16:(Redirected from
9140:
8928:Chiliagon (1000)
8908:Icositrigon (23)
8887:Octadecagon (18)
8877:Hexadecagon (16)
8781:
8780:
8600:
8593:
8586:
8577:
8576:
8564:Regular polytope
8125:
8114:
8103:
8062:
8005:
7998:
7991:
7982:
7981:
7977:
7976:
7949:
7948:
7920:
7914:
7913:
7893:
7887:
7886:
7870:
7860:
7854:
7853:
7829:
7823:
7822:
7800:
7791:H. S. M. Coxeter
7787:
7781:
7780:
7778:
7777:
7762:
7756:
7755:
7753:
7752:
7736:
7730:
7729:
7687:
7679:
7673:
7672:
7621:
7612:
7611:
7559:
7547:Grünbaum, Branko
7543:
7537:
7536:
7526:
7517:
7511:
7510:
7474:
7468:
7467:
7457:
7448:
7442:
7441:
7431:
7422:
7416:
7409:
7403:
7402:
7392:
7383:
7377:
7376:
7366:
7357:
7351:
7350:
7342:
7336:
7327:
7321:
7314:
7308:
7307:
7305:
7295:
7284:
7277:
7268:
7267:
7265:
7257:
7248:
7247:
7237:
7228:
7219:
7218:
7212:
7203:
7197:
7196:
7156:
7142:
7136:
7135:
7095:
7085:
7074:
7073:
7059:
7050:
7041:
7040:
6976:
6967:
6966:
6916:
6910:
6909:
6875:
6866:
6860:
6859:
6817:
6808:
6762:, including the
6727:
6725:
6724:
6719:
6697:
6695:
6694:
6689:
6666:
6664:
6663:
6658:
6571:hexagonal tiling
6554:
6552:
6551:
6546:
6538:
6537:
6528:
6527:
6515:
6514:
6499:
6498:
6482:
6480:
6479:
6474:
6462:
6460:
6459:
6454:
6452:
6451:
6435:
6433:
6432:
6427:
6425:
6424:
6408:
6406:
6405:
6400:
6398:
6397:
6375:
6373:
6372:
6367:
6362:
6354:
6349:
6347:
6346:
6337:
6336:
6327:
6322:
6314:
6302:
6300:
6299:
6294:
6292:
6291:
6275:
6273:
6272:
6267:
6265:
6264:
6243:
6241:
6240:
6235:
6230:
6228:
6227:
6222:
6213:
6199:
6197:
6196:
6191:
6189:
6187:
6186:
6181:
6172:
6152:
6150:
6149:
6144:
6139:
6131:
6126:
6118:
6113:
6105:
6096:
6094:
6093:
6088:
6070:
6068:
6067:
6062:
6060:
6057:
6056:
6055:
6043:
6042:
6032:
6031:
6030:
6018:
6017:
6007:
5997:
5995:
5994:
5989:
5984:
5982:
5971:
5970:
5969:
5948:
5947:
5937:
5922:
5920:
5919:
5914:
5902:
5900:
5899:
5894:
5879:
5877:
5876:
5871:
5859:
5857:
5856:
5851:
5836:
5834:
5833:
5828:
5813:
5811:
5810:
5805:
5790:
5788:
5787:
5782:
5767:
5765:
5764:
5759:
5744:
5742:
5741:
5736:
5722:
5720:
5719:
5714:
5709:
5708:
5696:
5695:
5674:
5673:
5658:
5656:
5655:
5650:
5623:
5621:
5620:
5615:
5603:
5601:
5600:
5595:
5583:
5581:
5580:
5575:
5563:
5561:
5560:
5555:
5543:
5541:
5540:
5535:
5523:
5521:
5520:
5515:
5503:
5501:
5500:
5495:
5480:
5478:
5477:
5472:
5458:
5456:
5455:
5450:
5445:
5444:
5429:
5425:
5424:
5423:
5411:
5410:
5398:
5397:
5374:
5372:
5371:
5366:
5361:
5360:
5345:
5341:
5340:
5339:
5327:
5326:
5314:
5313:
5289:
5287:
5286:
5281:
5269:
5267:
5266:
5261:
5249:
5247:
5246:
5241:
5229:
5227:
5226:
5221:
5206:
5204:
5203:
5198:
5186:
5184:
5183:
5178:
5166:
5164:
5163:
5158:
5146:
5144:
5143:
5138:
5133:
5129:
5128:
5127:
5118:
5117:
5102:
5101:
5096:
5092:
5091:
5090:
5078:
5077:
5051:
5050:
5038:
5037:
5025:
5024:
5009:
5007:
5006:
5001:
4996:
4992:
4991:
4990:
4978:
4977:
4957:
4956:
4944:
4943:
4931:
4930:
4914:
4912:
4911:
4906:
4894:
4892:
4891:
4886:
4874:
4872:
4871:
4866:
4854:
4852:
4851:
4846:
4832:
4830:
4829:
4824:
4819:
4818:
4813:
4809:
4808:
4807:
4795:
4794:
4782:
4781:
4769:
4768:
4750:
4746:
4745:
4744:
4732:
4731:
4719:
4718:
4706:
4705:
4681:
4679:
4678:
4673:
4661:
4659:
4658:
4653:
4641:
4639:
4638:
4633:
4621:
4619:
4618:
4613:
4601:
4599:
4598:
4593:
4581:
4579:
4578:
4573:
4561:
4559:
4558:
4553:
4534:
4532:
4531:
4526:
4514:
4512:
4511:
4506:
4494:
4492:
4491:
4486:
4474:
4472:
4471:
4466:
4445:
4443:
4442:
4437:
4416:
4414:
4413:
4408:
4383:
4381:
4380:
4375:
4355:
4353:
4352:
4347:
4335:
4333:
4332:
4327:
4313:
4311:
4310:
4305:
4297:
4289:
4280:
4278:
4277:
4272:
4261:to the inradius
4260:
4258:
4257:
4252:
4208:
4206:
4205:
4200:
4198:
4197:
4192:
4160:Other properties
4155:
4153:
4152:
4147:
4145:
4144:
4135:
4126:
4125:
4114:
4105:
4104:
4099:
4090:
4089:
4078:
4070:
4055:
4053:
4052:
4047:
4045:
4035:
4034:
4022:
4020:
4019:
4014:
4009:
4008:
3987:
3979:
3962:
3960:
3959:
3954:
3934:
3926:
3912:between them is
3911:
3909:
3908:
3903:
3891:
3889:
3888:
3883:
3871:
3869:
3868:
3863:
3840:
3838:
3837:
3832:
3827:
3826:
3817:
3808:
3807:
3792:
3790:
3789:
3784:
3776:
3768:
3759:
3757:
3756:
3751:
3737:
3735:
3734:
3729:
3721:
3712:
3711:
3696:
3694:
3693:
3688:
3686:
3685:
3673:
3672:
3660:
3659:
3654:
3650:
3642:
3625:
3623:
3622:
3617:
3605:
3603:
3602:
3597:
3581:
3579:
3578:
3567:
3565:
3564:
3550:
3548:
3547:
3542:
3531:
3523:
3509:from that side:
3508:
3506:
3505:
3500:
3488:
3486:
3485:
3480:
3460:
3458:
3457:
3452:
3440:
3438:
3437:
3432:
3430:
3429:
3420:
3411:
3410:
3363:
3361:
3360:
3355:
3312:
3310:
3309:
3304:
3288:
3286:
3285:
3280:
3265:
3263:
3262:
3257:
3242:
3240:
3239:
3234:
3219:
3217:
3216:
3211:
3196:
3194:
3193:
3188:
3173:
3171:
3170:
3165:
3146:
3144:
3143:
3138:
3120:
3118:
3117:
3112:
3101:states that, if
3094:
3092:
3091:
3086:
3074:
3072:
3071:
3066:
3033:
3031:
3030:
3025:
3013:
3011:
3010:
3005:
2993:
2991:
2990:
2985:
2973:
2971:
2970:
2965:
2953:
2951:
2950:
2945:
2928:is equilateral.
2904:angle trisectors
2891:
2889:
2888:
2883:
2865:
2863:
2862:
2857:
2839:
2837:
2836:
2831:
2804:
2802:
2801:
2796:
2778:
2776:
2775:
2770:
2752:
2750:
2749:
2744:
2726:
2724:
2723:
2718:
2703:
2701:
2700:
2695:
2680:
2678:
2677:
2672:
2657:
2655:
2654:
2649:
2634:
2632:
2631:
2626:
2611:
2609:
2608:
2603:
2585:
2583:
2582:
2577:
2548:Notable theorems
2543:
2541:
2540:
2535:
2530:
2529:
2517:
2516:
2504:
2503:
2491:
2487:
2486:
2485:
2473:
2472:
2460:
2459:
2435:
2433:
2432:
2427:
2415:
2413:
2412:
2407:
2395:
2393:
2392:
2387:
2375:
2373:
2372:
2367:
2355:
2353:
2352:
2347:
2335:
2333:
2332:
2327:
2315:
2313:
2312:
2307:
2179:
2177:
2176:
2171:
2169:
2168:
2156:
2155:
2143:
2142:
2125:
2123:
2122:
2117:
2115:
2110:
2109:
2108:
2096:
2095:
2083:
2082:
2072:
2056:
2054:
2053:
2048:
2046:
2045:
2033:
2032:
2020:
2019:
2007:
2006:
1985:
1983:
1982:
1977:
1948:
1946:
1945:
1940:
1938:
1937:
1929:
1911:
1902:
1901:
1880:
1878:
1877:
1872:
1869:
1867:
1866:
1861:
1855:
1854:
1853:
1841:
1840:
1828:
1827:
1817:
1794:
1792:
1791:
1786:
1784:
1776:
1771:
1763:
1755:
1747:
1739:
1731:
1715:
1713:
1712:
1707:
1705:
1697:
1692:
1678:
1664:
1645:
1643:
1642:
1637:
1635:
1634:
1592:
1590:
1589:
1584:
1579:
1574:
1573:
1568:
1562:
1546:
1544:
1543:
1538:
1533:
1528:
1509:
1507:
1506:
1501:
1496:
1491:
1483:
1482:
1465:
1463:
1462:
1457:
1443:
1442:
1427:
1426:
1408:
1406:
1405:
1400:
1395:
1391:
1384:
1379:
1339:
1337:
1336:
1331:
1329:
1327:
1315:
1314:
1290:
1289:
1284:
1276:
1271:
1263:
1258:
1250:
1240:
1238:
1237:
1232:
1187:
1185:
1184:
1179:
1167:
1165:
1164:
1159:
1147:
1145:
1144:
1139:
1127:
1125:
1124:
1119:
1107:
1105:
1104:
1099:
1087:
1085:
1084:
1079:
1077:
1076:
1060:
1058:
1057:
1052:
1050:
1049:
1033:
1031:
1030:
1025:
1023:
1022:
1003:
1001:
1000:
995:
980:
978:
977:
972:
957:
955:
954:
949:
937:
935:
934:
929:
917:
915:
914:
909:
897:
895:
894:
889:
860:
858:
857:
852:
850:
842:
825:
823:
822:
817:
815:
810:
802:
785:
783:
782:
777:
775:
767:
752:
750:
749:
744:
742:
736:
735:
734:
725:
703:
701:
700:
695:
693:
692:
683:
678:
677:
672:
666:
636:
634:
633:
628:
623:
614:
613:
589:
587:
586:
581:
579:
571:
556:
554:
553:
548:
543:
534:
533:
517:inscribed circle
512:
510:
509:
504:
502:
496:
492:
471:
469:
468:
463:
438:
436:
435:
430:
425:
424:
415:
406:
405:
382:
380:
379:
374:
358:
356:
355:
350:
348:
347:
335:
334:
322:
321:
305:
303:
302:
297:
273:
271:
270:
265:
228:regular triangle
173:
171:
170:
165:
163:
162:
153:
143:
142:
109:
108:
107:
103:
102:
98:
97:
49:
37:
36:
21:
18:Regular triangle
9148:
9147:
9143:
9142:
9141:
9139:
9138:
9137:
9118:
9117:
9116:
9111:
9010:
8964:
8952:
8896:
8862:Tridecagon (13)
8852:Hendecagon (11)
8840:
8776:
8770:
8741:Right trapezoid
8662:
8614:
8604:
8574:
8543:
8536:
8529:
8412:
8405:
8398:
8362:
8355:
8348:
8312:
8305:
8139:Regular polygon
8132:
8123:
8116:
8112:
8105:
8101:
8092:
8083:
8076:
8072:
8060:
8054:
8050:
8038:
8020:
8009:
7958:
7953:
7952:
7921:
7917:
7910:
7894:
7890:
7883:
7861:
7857:
7850:
7830:
7826:
7788:
7784:
7775:
7773:
7763:
7759:
7750:
7748:
7737:
7733:
7702:
7680:
7676:
7653:
7622:
7615:
7576:10.2307/2689529
7557:
7544:
7540:
7524:
7518:
7514:
7475:
7471:
7455:
7449:
7445:
7429:
7423:
7419:
7410:
7406:
7390:
7384:
7380:
7364:
7358:
7354:
7343:
7339:
7328:
7324:
7315:
7311:
7296:
7287:
7278:
7271:
7263:
7259:
7258:
7251:
7235:
7229:
7222:
7217:(Course Notes).
7210:
7204:
7200:
7177:
7143:
7139:
7116:
7086:
7077:
7057:
7051:
7044:
7005:
6977:
6970:
6939:10.2307/2687913
6917:
6913:
6873:
6867:
6863:
6815:
6809:
6805:
6800:
6773:
6734:
6707:
6704:
6703:
6683:
6680:
6679:
6649:
6646:
6645:
6638:star antiprisms
6614:Platonic solids
6533:
6529:
6523:
6519:
6510:
6506:
6494:
6490:
6488:
6485:
6484:
6468:
6465:
6464:
6447:
6443:
6441:
6438:
6437:
6420:
6416:
6414:
6411:
6410:
6393:
6389:
6387:
6384:
6383:
6353:
6342:
6338:
6332:
6328:
6326:
6313:
6311:
6308:
6307:
6287:
6283:
6281:
6278:
6277:
6260:
6256:
6254:
6251:
6250:
6221:
6217:
6212:
6210:
6207:
6206:
6180:
6176:
6171:
6169:
6166:
6165:
6130:
6117:
6104:
6102:
6099:
6098:
6076:
6073:
6072:
6051:
6047:
6038:
6034:
6033:
6026:
6022:
6013:
6009:
6008:
6005:
6003:
6000:
5999:
5972:
5965:
5961:
5943:
5939:
5938:
5936:
5928:
5925:
5924:
5908:
5905:
5904:
5885:
5882:
5881:
5865:
5862:
5861:
5842:
5839:
5838:
5819:
5816:
5815:
5796:
5793:
5792:
5773:
5770:
5769:
5750:
5747:
5746:
5730:
5727:
5726:
5704:
5700:
5691:
5687:
5669:
5665:
5663:
5660:
5659:
5629:
5626:
5625:
5624:, respectively
5609:
5606:
5605:
5589:
5586:
5585:
5569:
5566:
5565:
5549:
5546:
5545:
5529:
5526:
5525:
5509:
5506:
5505:
5486:
5483:
5482:
5466:
5463:
5462:
5440:
5436:
5419:
5415:
5406:
5402:
5393:
5389:
5388:
5384:
5379:
5376:
5375:
5356:
5352:
5335:
5331:
5322:
5318:
5309:
5305:
5304:
5300:
5295:
5292:
5291:
5275:
5272:
5271:
5255:
5252:
5251:
5235:
5232:
5231:
5215:
5212:
5211:
5192:
5189:
5188:
5172:
5169:
5168:
5152:
5149:
5148:
5123:
5119:
5113:
5109:
5097:
5086:
5082:
5073:
5069:
5068:
5064:
5063:
5062:
5058:
5046:
5042:
5033:
5029:
5020:
5016:
5014:
5011:
5010:
4986:
4982:
4973:
4969:
4968:
4964:
4952:
4948:
4939:
4935:
4926:
4922:
4920:
4917:
4916:
4900:
4897:
4896:
4880:
4877:
4876:
4860:
4857:
4856:
4840:
4837:
4836:
4814:
4803:
4799:
4790:
4786:
4777:
4773:
4764:
4760:
4759:
4755:
4754:
4740:
4736:
4727:
4723:
4714:
4710:
4701:
4697:
4696:
4692:
4687:
4684:
4683:
4667:
4664:
4663:
4647:
4644:
4643:
4627:
4624:
4623:
4607:
4604:
4603:
4587:
4584:
4583:
4567:
4564:
4563:
4547:
4544:
4543:
4520:
4517:
4516:
4500:
4497:
4496:
4480:
4477:
4476:
4451:
4448:
4447:
4422:
4419:
4418:
4393:
4390:
4389:
4386:angle bisectors
4369:
4366:
4365:
4341:
4338:
4337:
4321:
4318:
4317:
4288:
4286:
4283:
4282:
4266:
4263:
4262:
4246:
4243:
4242:
4215:orthic triangle
4193:
4188:
4187:
4185:
4182:
4181:
4162:
4140:
4136:
4124:
4103:
4088:
4069:
4061:
4058:
4057:
4032:
4030:
4027:
4026:
4004:
4000:
3978:
3970:
3967:
3966:
3925:
3917:
3914:
3913:
3897:
3894:
3893:
3892:, and an angle
3877:
3874:
3873:
3857:
3854:
3853:
3846:
3822:
3818:
3806:
3798:
3795:
3794:
3767:
3765:
3762:
3761:
3745:
3742:
3741:
3710:
3702:
3699:
3698:
3681:
3677:
3668:
3664:
3655:
3641:
3637:
3636:
3634:
3631:
3630:
3611:
3608:
3607:
3591:
3588:
3587:
3576:
3574:
3573:
3562:
3560:
3558:
3522:
3514:
3511:
3510:
3494:
3491:
3490:
3474:
3471:
3470:
3467:
3446:
3443:
3442:
3425:
3421:
3409:
3401:
3398:
3397:
3394:
3349:
3346:
3345:
3323:
3298:
3295:
3294:
3271:
3268:
3267:
3248:
3245:
3244:
3225:
3222:
3221:
3202:
3199:
3198:
3179:
3176:
3175:
3156:
3153:
3152:
3147:but not on its
3126:
3123:
3122:
3106:
3103:
3102:
3080:
3077:
3076:
3039:
3036:
3035:
3019:
3016:
3015:
2999:
2996:
2995:
2979:
2976:
2975:
2959:
2956:
2955:
2939:
2936:
2935:
2895:
2871:
2868:
2867:
2845:
2842:
2841:
2816:
2813:
2812:
2784:
2781:
2780:
2758:
2755:
2754:
2732:
2729:
2728:
2709:
2706:
2705:
2686:
2683:
2682:
2663:
2660:
2659:
2640:
2637:
2636:
2617:
2614:
2613:
2594:
2591:
2590:
2565:
2562:
2561:
2550:
2525:
2521:
2512:
2508:
2499:
2495:
2481:
2477:
2468:
2464:
2455:
2451:
2450:
2446:
2441:
2438:
2437:
2421:
2418:
2417:
2401:
2398:
2397:
2381:
2378:
2377:
2361:
2358:
2357:
2341:
2338:
2337:
2321:
2318:
2317:
2301:
2298:
2297:
2289:
2269:
2247:, centroid, or
2225:triangle center
2221:
2212:angle bisectors
2188:Three kinds of
2186:
2164:
2160:
2151:
2147:
2138:
2134:
2132:
2129:
2128:
2104:
2100:
2091:
2087:
2078:
2074:
2073:
2071:
2063:
2060:
2059:
2041:
2037:
2028:
2024:
2015:
2011:
2002:
1998:
1993:
1990:
1989:
1986:(Chapple-Euler)
1962:
1959:
1958:
1955:
1928:
1924:
1900:
1892:
1889:
1888:
1860:
1856:
1849:
1845:
1836:
1832:
1823:
1819:
1818:
1816:
1808:
1805:
1804:
1801:
1775:
1762:
1746:
1730:
1722:
1719:
1718:
1696:
1688:
1674:
1660:
1652:
1649:
1648:
1630:
1626:
1606:
1603:
1602:
1599:
1567:
1563:
1561:
1553:
1550:
1549:
1527:
1516:
1513:
1512:
1490:
1478:
1474:
1472:
1469:
1468:
1438:
1434:
1422:
1418:
1416:
1413:
1412:
1378:
1374:
1370:
1353:
1350:
1349:
1346:
1317:
1310:
1306:
1288:
1275:
1262:
1249:
1247:
1244:
1243:
1214:
1211:
1210:
1207:
1173:
1170:
1169:
1153:
1150:
1149:
1133:
1130:
1129:
1113:
1110:
1109:
1093:
1090:
1089:
1072:
1068:
1066:
1063:
1062:
1045:
1041:
1039:
1036:
1035:
1018:
1014:
1012:
1009:
1008:
989:
986:
985:
966:
963:
962:
943:
940:
939:
923:
920:
919:
903:
900:
899:
877:
874:
873:
870:
841:
833:
830:
829:
803:
801:
793:
790:
789:
766:
764:
761:
760:
730:
726:
724:
716:
713:
712:
688:
684:
671:
667:
665:
657:
654:
653:
612:
604:
601:
600:
570:
562:
559:
558:
532:
524:
521:
520:
491:
483:
480:
479:
446:
443:
442:
420:
416:
404:
396:
393:
392:
368:
365:
364:
343:
339:
330:
326:
317:
313:
311:
308:
307:
279:
276:
275:
247:
244:
243:
236:
224:regular polygon
158:
154:
140:
138:
135:
134:
123:
105:
100:
95:
93:
79:Schläfli symbol
58:Regular polygon
35:
28:
23:
22:
15:
12:
11:
5:
9146:
9136:
9135:
9130:
9113:
9112:
9110:
9109:
9104:
9099:
9094:
9089:
9084:
9079:
9074:
9069:
9067:Pseudotriangle
9064:
9059:
9054:
9049:
9044:
9039:
9034:
9029:
9024:
9018:
9016:
9012:
9011:
9009:
9008:
9003:
8998:
8993:
8988:
8983:
8978:
8973:
8967:
8965:
8958:
8957:
8954:
8953:
8951:
8950:
8945:
8940:
8935:
8930:
8925:
8920:
8915:
8910:
8904:
8902:
8898:
8897:
8895:
8894:
8889:
8884:
8879:
8874:
8869:
8864:
8859:
8857:Dodecagon (12)
8854:
8848:
8846:
8842:
8841:
8839:
8838:
8833:
8828:
8823:
8818:
8813:
8808:
8803:
8798:
8793:
8787:
8785:
8778:
8772:
8771:
8769:
8768:
8763:
8758:
8753:
8748:
8743:
8738:
8733:
8728:
8723:
8718:
8713:
8708:
8703:
8698:
8693:
8688:
8683:
8678:
8672:
8670:
8668:Quadrilaterals
8664:
8663:
8661:
8660:
8655:
8650:
8645:
8640:
8635:
8630:
8624:
8622:
8616:
8615:
8603:
8602:
8595:
8588:
8580:
8571:
8570:
8555:
8554:
8545:
8541:
8534:
8527:
8523:
8514:
8497:
8488:
8477:
8476:
8474:
8472:
8467:
8458:
8453:
8447:
8446:
8444:
8442:
8437:
8428:
8423:
8417:
8416:
8414:
8410:
8403:
8396:
8392:
8387:
8378:
8373:
8367:
8366:
8364:
8360:
8353:
8346:
8342:
8337:
8328:
8323:
8317:
8316:
8314:
8310:
8303:
8299:
8294:
8285:
8280:
8274:
8273:
8271:
8269:
8264:
8255:
8250:
8244:
8243:
8234:
8229:
8224:
8215:
8210:
8204:
8203:
8194:
8192:
8187:
8178:
8173:
8167:
8166:
8161:
8156:
8151:
8146:
8141:
8135:
8134:
8130:
8126:
8121:
8110:
8099:
8090:
8081:
8074:
8068:
8058:
8052:
8046:
8040:
8034:
8028:
8022:
8021:
8010:
8008:
8007:
8000:
7993:
7985:
7980:
7979:
7978:
7957:
7956:External links
7954:
7951:
7950:
7931:(6): 737–742.
7915:
7909:978-0810872462
7908:
7888:
7882:978-0313339943
7881:
7855:
7849:978-0972488129
7848:
7824:
7782:
7765:Webb, Robert.
7757:
7731:
7701:978-0521664059
7700:
7674:
7652:978-1107103405
7651:
7613:
7538:
7512:
7469:
7443:
7417:
7415:, 1979, p. 65.
7404:
7378:
7352:
7337:
7322:
7309:
7285:
7269:
7249:
7220:
7198:
7175:
7137:
7114:
7075:
7064:(3): 121–123.
7042:
7003:
6968:
6911:
6861:
6802:
6801:
6799:
6796:
6795:
6794:
6789:
6784:
6779:
6772:
6769:
6768:
6767:
6756:
6745:
6733:
6730:
6717:
6714:
6711:
6687:
6656:
6653:
6544:
6541:
6536:
6532:
6526:
6522:
6518:
6513:
6509:
6505:
6502:
6497:
6493:
6472:
6450:
6446:
6423:
6419:
6396:
6392:
6365:
6360:
6357:
6352:
6345:
6341:
6335:
6331:
6325:
6320:
6317:
6305:
6304:
6290:
6286:
6263:
6259:
6246:
6245:
6233:
6225:
6220:
6216:
6202:
6201:
6184:
6179:
6175:
6155:optic equation
6142:
6137:
6134:
6129:
6124:
6121:
6116:
6111:
6108:
6086:
6083:
6080:
6054:
6050:
6046:
6041:
6037:
6029:
6025:
6021:
6016:
6012:
5987:
5981:
5978:
5975:
5968:
5964:
5960:
5957:
5954:
5951:
5946:
5942:
5935:
5932:
5912:
5903:having length
5892:
5889:
5869:
5860:having length
5849:
5846:
5826:
5823:
5803:
5800:
5791:into segments
5780:
5777:
5757:
5754:
5734:
5712:
5707:
5703:
5699:
5694:
5690:
5686:
5683:
5680:
5677:
5672:
5668:
5648:
5645:
5642:
5639:
5636:
5633:
5613:
5593:
5573:
5553:
5533:
5513:
5493:
5490:
5470:
5461:For any point
5448:
5443:
5439:
5435:
5432:
5428:
5422:
5418:
5414:
5409:
5405:
5401:
5396:
5392:
5387:
5383:
5364:
5359:
5355:
5351:
5348:
5344:
5338:
5334:
5330:
5325:
5321:
5317:
5312:
5308:
5303:
5299:
5279:
5259:
5239:
5219:
5210:For any point
5196:
5176:
5156:
5136:
5132:
5126:
5122:
5116:
5112:
5108:
5105:
5100:
5095:
5089:
5085:
5081:
5076:
5072:
5067:
5061:
5057:
5054:
5049:
5045:
5041:
5036:
5032:
5028:
5023:
5019:
4999:
4995:
4989:
4985:
4981:
4976:
4972:
4967:
4963:
4960:
4955:
4951:
4947:
4942:
4938:
4934:
4929:
4925:
4904:
4884:
4864:
4844:
4835:For any point
4822:
4817:
4812:
4806:
4802:
4798:
4793:
4789:
4785:
4780:
4776:
4772:
4767:
4763:
4758:
4753:
4749:
4743:
4739:
4735:
4730:
4726:
4722:
4717:
4713:
4709:
4704:
4700:
4695:
4691:
4682:respectively,
4671:
4651:
4631:
4611:
4591:
4571:
4551:
4542:For any point
4524:
4504:
4484:
4464:
4461:
4458:
4455:
4435:
4432:
4429:
4426:
4406:
4403:
4400:
4397:
4373:
4345:
4325:
4316:Given a point
4303:
4300:
4295:
4292:
4270:
4250:
4196:
4191:
4174:symmetry group
4161:
4158:
4143:
4139:
4133:
4129:
4123:
4120:
4117:
4112:
4108:
4102:
4097:
4093:
4087:
4084:
4081:
4076:
4073:
4068:
4065:
4042:
4038:
4012:
4007:
4003:
3999:
3996:
3993:
3990:
3985:
3982:
3977:
3974:
3952:
3949:
3946:
3943:
3940:
3937:
3932:
3929:
3924:
3921:
3901:
3881:
3861:
3845:
3842:
3830:
3825:
3821:
3815:
3811:
3805:
3802:
3782:
3779:
3774:
3771:
3749:
3727:
3724:
3719:
3715:
3709:
3706:
3684:
3680:
3676:
3671:
3667:
3663:
3658:
3653:
3648:
3645:
3640:
3615:
3595:
3540:
3537:
3534:
3529:
3526:
3521:
3518:
3498:
3478:
3466:
3463:
3450:
3428:
3424:
3418:
3414:
3408:
3405:
3393:
3390:
3353:
3322:
3319:
3302:
3278:
3275:
3255:
3252:
3232:
3229:
3209:
3206:
3186:
3183:
3163:
3160:
3136:
3133:
3130:
3110:
3084:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3023:
3003:
2983:
2963:
2943:
2894:
2893:
2881:
2878:
2875:
2855:
2852:
2849:
2829:
2826:
2823:
2820:
2809:
2806:
2794:
2791:
2788:
2768:
2765:
2762:
2742:
2739:
2736:
2716:
2713:
2693:
2690:
2670:
2667:
2647:
2644:
2624:
2621:
2601:
2598:
2587:
2575:
2572:
2569:
2557:
2549:
2546:
2545:
2544:
2533:
2528:
2524:
2520:
2515:
2511:
2507:
2502:
2498:
2494:
2490:
2484:
2480:
2476:
2471:
2467:
2463:
2458:
2454:
2449:
2445:
2425:
2405:
2385:
2365:
2345:
2325:
2305:
2288:
2285:
2284:
2283:
2280:
2268:
2265:
2264:
2263:
2252:
2220:
2217:
2216:
2215:
2208:
2201:
2185:
2182:
2181:
2180:
2167:
2163:
2159:
2154:
2150:
2146:
2141:
2137:
2126:
2113:
2107:
2103:
2099:
2094:
2090:
2086:
2081:
2077:
2070:
2067:
2057:
2044:
2040:
2036:
2031:
2027:
2023:
2018:
2014:
2010:
2005:
2001:
1997:
1987:
1975:
1972:
1969:
1966:
1954:
1951:
1950:
1949:
1935:
1932:
1927:
1923:
1920:
1917:
1914:
1909:
1905:
1899:
1896:
1886:
1864:
1859:
1852:
1848:
1844:
1839:
1835:
1831:
1826:
1822:
1815:
1812:
1800:
1797:
1796:
1795:
1782:
1779:
1774:
1769:
1766:
1761:
1758:
1753:
1750:
1745:
1742:
1737:
1734:
1729:
1726:
1716:
1703:
1700:
1695:
1691:
1687:
1684:
1681:
1677:
1673:
1670:
1667:
1663:
1659:
1656:
1646:
1633:
1629:
1625:
1622:
1619:
1616:
1613:
1610:
1598:
1595:
1594:
1593:
1582:
1577:
1571:
1566:
1560:
1557:
1547:
1536:
1531:
1526:
1523:
1520:
1510:
1499:
1494:
1489:
1486:
1481:
1477:
1466:
1455:
1452:
1449:
1446:
1441:
1437:
1433:
1430:
1425:
1421:
1410:
1398:
1394:
1390:
1387:
1382:
1377:
1373:
1369:
1366:
1363:
1360:
1357:
1345:
1342:
1341:
1340:
1326:
1323:
1320:
1313:
1309:
1305:
1302:
1299:
1296:
1293:
1287:
1282:
1279:
1274:
1269:
1266:
1261:
1256:
1253:
1241:
1230:
1227:
1224:
1221:
1218:
1206:
1203:
1198:if and only if
1177:
1157:
1137:
1117:
1097:
1075:
1071:
1048:
1044:
1021:
1017:
993:
970:
947:
927:
907:
887:
884:
881:
869:
866:
862:
861:
848:
845:
840:
837:
826:
813:
809:
806:
800:
797:
786:
773:
770:
753:
739:
733:
729:
723:
720:
705:
704:
691:
687:
681:
675:
670:
664:
661:
638:
637:
626:
621:
617:
611:
608:
593:
590:
577:
574:
569:
566:
546:
541:
537:
531:
528:
513:
499:
495:
490:
487:
472:
459:
456:
453:
450:
439:
428:
423:
419:
413:
409:
403:
400:
372:
346:
342:
338:
333:
329:
325:
320:
316:
295:
292:
289:
286:
283:
263:
260:
257:
254:
251:
235:
232:
190:
189:
186:
179:Internal angle
175:
174:
161:
157:
150:
146:
132:
126:
125:
121:
117:
115:Symmetry group
111:
110:
91:
85:
84:
81:
75:
74:
71:
61:
60:
55:
51:
50:
42:
41:
26:
9:
6:
4:
3:
2:
9145:
9134:
9131:
9129:
9126:
9125:
9123:
9108:
9107:Weakly simple
9105:
9103:
9100:
9098:
9095:
9093:
9090:
9088:
9085:
9083:
9080:
9078:
9075:
9073:
9070:
9068:
9065:
9063:
9060:
9058:
9055:
9053:
9050:
9048:
9047:Infinite skew
9045:
9043:
9040:
9038:
9035:
9033:
9030:
9028:
9025:
9023:
9020:
9019:
9017:
9013:
9007:
9004:
9002:
8999:
8997:
8994:
8992:
8989:
8987:
8984:
8982:
8979:
8977:
8974:
8972:
8969:
8968:
8966:
8963:
8962:Star polygons
8959:
8949:
8948:Apeirogon (∞)
8946:
8944:
8941:
8939:
8936:
8934:
8931:
8929:
8926:
8924:
8921:
8919:
8916:
8914:
8911:
8909:
8906:
8905:
8903:
8899:
8893:
8892:Icosagon (20)
8890:
8888:
8885:
8883:
8880:
8878:
8875:
8873:
8870:
8868:
8865:
8863:
8860:
8858:
8855:
8853:
8850:
8849:
8847:
8843:
8837:
8834:
8832:
8829:
8827:
8824:
8822:
8819:
8817:
8814:
8812:
8809:
8807:
8804:
8802:
8799:
8797:
8794:
8792:
8789:
8788:
8786:
8782:
8779:
8773:
8767:
8764:
8762:
8759:
8757:
8754:
8752:
8749:
8747:
8744:
8742:
8739:
8737:
8734:
8732:
8729:
8727:
8726:Parallelogram
8724:
8722:
8721:Orthodiagonal
8719:
8717:
8714:
8712:
8709:
8707:
8704:
8702:
8701:Ex-tangential
8699:
8697:
8694:
8692:
8689:
8687:
8684:
8682:
8679:
8677:
8674:
8673:
8671:
8669:
8665:
8659:
8656:
8654:
8651:
8649:
8646:
8644:
8641:
8639:
8636:
8634:
8631:
8629:
8626:
8625:
8623:
8621:
8617:
8612:
8608:
8601:
8596:
8594:
8589:
8587:
8582:
8581:
8578:
8569:
8565:
8561:
8556:
8553:
8549:
8546:
8544:
8537:
8530:
8524:
8522:
8518:
8515:
8513:
8509:
8505:
8501:
8498:
8496:
8492:
8489:
8487:
8483:
8479:
8478:
8475:
8473:
8471:
8468:
8466:
8462:
8459:
8457:
8454:
8452:
8449:
8448:
8445:
8443:
8441:
8438:
8436:
8432:
8429:
8427:
8424:
8422:
8419:
8418:
8415:
8413:
8406:
8399:
8393:
8391:
8388:
8386:
8382:
8379:
8377:
8374:
8372:
8369:
8368:
8365:
8363:
8356:
8349:
8343:
8341:
8338:
8336:
8332:
8329:
8327:
8324:
8322:
8319:
8318:
8315:
8313:
8306:
8300:
8298:
8295:
8293:
8289:
8286:
8284:
8281:
8279:
8276:
8275:
8272:
8270:
8268:
8265:
8263:
8259:
8256:
8254:
8251:
8249:
8246:
8245:
8242:
8238:
8235:
8233:
8230:
8228:
8227:Demitesseract
8225:
8223:
8219:
8216:
8214:
8211:
8209:
8206:
8205:
8202:
8198:
8195:
8193:
8191:
8188:
8186:
8182:
8179:
8177:
8174:
8172:
8169:
8168:
8165:
8162:
8160:
8157:
8155:
8152:
8150:
8147:
8145:
8142:
8140:
8137:
8136:
8133:
8127:
8124:
8120:
8113:
8109:
8102:
8098:
8093:
8089:
8084:
8080:
8075:
8073:
8071:
8067:
8057:
8053:
8051:
8049:
8045:
8041:
8039:
8037:
8033:
8029:
8027:
8024:
8023:
8018:
8014:
8006:
8001:
7999:
7994:
7992:
7987:
7986:
7983:
7974:
7973:
7968:
7965:
7960:
7959:
7946:
7942:
7938:
7934:
7930:
7926:
7919:
7911:
7905:
7901:
7900:
7892:
7884:
7878:
7874:
7869:
7868:
7859:
7851:
7845:
7841:
7840:160, 224, 226
7837:
7836:
7828:
7820:
7816:
7812:
7808:
7804:
7799:
7798:
7792:
7786:
7772:
7768:
7761:
7746:
7742:
7735:
7727:
7723:
7719:
7715:
7711:
7707:
7703:
7697:
7693:
7692:
7686:
7678:
7670:
7666:
7662:
7658:
7654:
7648:
7644:
7640:
7636:
7632:
7631:
7626:
7620:
7618:
7609:
7605:
7601:
7597:
7593:
7589:
7585:
7581:
7577:
7573:
7569:
7565:
7564:
7556:
7552:
7548:
7542:
7534:
7530:
7523:
7516:
7508:
7504:
7500:
7496:
7492:
7488:
7484:
7480:
7473:
7465:
7461:
7454:
7447:
7439:
7435:
7428:
7421:
7414:
7408:
7400:
7396:
7389:
7382:
7374:
7370:
7363:
7356:
7348:
7341:
7334:
7333:
7326:
7319:
7313:
7306:. Dover Publ.
7304:
7303:
7294:
7292:
7290:
7282:
7276:
7274:
7262:
7256:
7254:
7245:
7241:
7234:
7227:
7225:
7216:
7209:
7202:
7194:
7190:
7186:
7182:
7178:
7176:9780883857632
7172:
7168:
7164:
7160:
7155:
7154:
7148:
7141:
7133:
7129:
7125:
7121:
7117:
7111:
7107:
7103:
7099:
7094:
7093:
7084:
7082:
7080:
7071:
7067:
7063:
7056:
7049:
7047:
7038:
7034:
7030:
7026:
7022:
7018:
7014:
7010:
7006:
7000:
6996:
6992:
6988:
6984:
6983:
6975:
6973:
6964:
6960:
6956:
6952:
6948:
6944:
6940:
6936:
6932:
6928:
6924:
6923:
6915:
6907:
6903:
6899:
6895:
6891:
6887:
6883:
6879:
6872:
6865:
6857:
6853:
6849:
6845:
6841:
6837:
6833:
6829:
6825:
6821:
6814:
6807:
6803:
6793:
6790:
6788:
6785:
6783:
6780:
6778:
6775:
6774:
6765:
6761:
6757:
6754:
6750:
6746:
6743:
6739:
6738:
6737:
6729:
6715:
6712:
6709:
6701:
6685:
6676:
6674:
6670:
6669:star polygons
6654:
6651:
6643:
6639:
6635:
6631:
6627:
6623:
6619:
6615:
6611:
6602:
6598:
6596:
6592:
6588:
6584:
6580:
6576:
6572:
6568:
6559:
6555:
6542:
6539:
6534:
6530:
6524:
6520:
6516:
6511:
6507:
6503:
6500:
6495:
6491:
6470:
6448:
6444:
6421:
6417:
6394:
6390:
6381:
6380:complex plane
6376:
6363:
6358:
6355:
6350:
6343:
6339:
6333:
6329:
6323:
6318:
6315:
6288:
6284:
6261:
6257:
6248:
6247:
6231:
6223:
6218:
6214:
6204:
6203:
6182:
6177:
6173:
6163:
6162:
6161:
6158:
6156:
6153:which is the
6140:
6135:
6132:
6127:
6122:
6119:
6114:
6109:
6106:
6084:
6081:
6078:
6052:
6048:
6044:
6039:
6035:
6027:
6023:
6019:
6014:
6010:
5985:
5979:
5976:
5973:
5966:
5962:
5958:
5955:
5952:
5949:
5944:
5940:
5933:
5930:
5910:
5890:
5887:
5867:
5847:
5844:
5824:
5821:
5801:
5798:
5778:
5775:
5755:
5752:
5732:
5723:
5710:
5705:
5701:
5697:
5692:
5688:
5684:
5681:
5678:
5675:
5670:
5666:
5646:
5643:
5640:
5637:
5634:
5631:
5611:
5591:
5571:
5551:
5531:
5511:
5491:
5488:
5468:
5459:
5446:
5441:
5437:
5433:
5430:
5426:
5420:
5416:
5412:
5407:
5403:
5399:
5394:
5390:
5385:
5381:
5362:
5357:
5353:
5349:
5346:
5342:
5336:
5332:
5328:
5323:
5319:
5315:
5310:
5306:
5301:
5297:
5277:
5257:
5237:
5217:
5208:
5194:
5174:
5154:
5134:
5130:
5124:
5120:
5114:
5110:
5106:
5103:
5098:
5093:
5087:
5083:
5079:
5074:
5070:
5065:
5059:
5055:
5052:
5047:
5043:
5039:
5034:
5030:
5026:
5021:
5017:
4997:
4993:
4987:
4983:
4979:
4974:
4970:
4965:
4961:
4958:
4953:
4949:
4945:
4940:
4936:
4932:
4927:
4923:
4902:
4882:
4862:
4842:
4833:
4820:
4815:
4810:
4804:
4800:
4796:
4791:
4787:
4783:
4778:
4774:
4770:
4765:
4761:
4756:
4751:
4747:
4741:
4737:
4733:
4728:
4724:
4720:
4715:
4711:
4707:
4702:
4698:
4693:
4689:
4669:
4649:
4629:
4609:
4589:
4569:
4549:
4540:
4538:
4522:
4502:
4482:
4462:
4459:
4456:
4433:
4430:
4427:
4404:
4401:
4398:
4387:
4371:
4363:
4359:
4343:
4323:
4314:
4301:
4298:
4293:
4290:
4268:
4248:
4240:
4235:
4233:
4229:
4224:
4220:
4216:
4212:
4194:
4179:
4175:
4171:
4167:
4157:
4141:
4137:
4131:
4127:
4121:
4118:
4115:
4110:
4106:
4100:
4095:
4091:
4085:
4082:
4079:
4074:
4071:
4066:
4063:
4040:
4036:
4023:
4010:
4005:
4001:
3997:
3994:
3991:
3988:
3983:
3980:
3975:
3972:
3963:
3950:
3947:
3944:
3941:
3938:
3935:
3930:
3927:
3922:
3919:
3899:
3879:
3859:
3851:
3841:
3828:
3823:
3819:
3813:
3809:
3803:
3800:
3780:
3777:
3772:
3769:
3747:
3740:Substituting
3738:
3725:
3722:
3717:
3713:
3707:
3704:
3682:
3678:
3674:
3669:
3665:
3661:
3656:
3651:
3646:
3643:
3638:
3629:
3613:
3593:
3571:
3566:
3555:
3551:
3538:
3535:
3532:
3527:
3524:
3519:
3516:
3496:
3476:
3462:
3448:
3426:
3422:
3416:
3412:
3406:
3403:
3386:
3382:
3380:
3379:
3372:
3370:
3369:vesica piscis
3365:
3351:
3342:
3340:
3336:
3327:
3318:
3316:
3300:
3292:
3276:
3273:
3253:
3250:
3230:
3227:
3207:
3204:
3184:
3181:
3161:
3158:
3150:
3134:
3131:
3128:
3108:
3100:
3096:
3082:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3021:
3001:
2981:
2961:
2941:
2933:
2929:
2927:
2923:
2919:
2914:
2911:
2907:
2905:
2901:
2879:
2876:
2873:
2853:
2850:
2847:
2827:
2824:
2821:
2818:
2810:
2807:
2792:
2789:
2786:
2766:
2763:
2760:
2740:
2737:
2734:
2714:
2711:
2691:
2688:
2668:
2665:
2645:
2642:
2622:
2619:
2599:
2596:
2588:
2573:
2570:
2567:
2559:
2558:
2554:
2531:
2526:
2522:
2518:
2513:
2509:
2505:
2500:
2496:
2492:
2488:
2482:
2478:
2474:
2469:
2465:
2461:
2456:
2452:
2447:
2443:
2423:
2403:
2383:
2363:
2343:
2323:
2303:
2295:
2291:
2290:
2281:
2278:
2277:
2276:
2274:
2261:
2257:
2253:
2250:
2246:
2242:
2238:
2237:
2236:
2234:
2230:
2226:
2213:
2209:
2206:
2202:
2199:
2195:
2194:
2193:
2191:
2184:Equal cevians
2165:
2161:
2157:
2152:
2148:
2144:
2139:
2135:
2127:
2111:
2105:
2101:
2097:
2092:
2088:
2084:
2079:
2075:
2068:
2065:
2058:
2042:
2038:
2034:
2029:
2025:
2021:
2016:
2012:
2008:
2003:
1999:
1995:
1988:
1973:
1970:
1967:
1964:
1957:
1956:
1933:
1930:
1921:
1918:
1915:
1907:
1903:
1897:
1894:
1887:
1884:
1862:
1857:
1850:
1846:
1842:
1837:
1833:
1829:
1824:
1820:
1813:
1810:
1803:
1802:
1780:
1777:
1772:
1767:
1764:
1759:
1756:
1751:
1748:
1743:
1740:
1735:
1732:
1727:
1724:
1717:
1701:
1698:
1693:
1689:
1685:
1682:
1679:
1675:
1671:
1668:
1665:
1661:
1657:
1654:
1647:
1631:
1627:
1623:
1620:
1617:
1614:
1611:
1608:
1601:
1600:
1580:
1575:
1569:
1564:
1558:
1555:
1548:
1534:
1529:
1524:
1521:
1518:
1511:
1497:
1492:
1487:
1484:
1479:
1475:
1467:
1453:
1450:
1447:
1444:
1439:
1435:
1431:
1428:
1423:
1419:
1411:
1396:
1392:
1388:
1385:
1380:
1375:
1371:
1367:
1364:
1361:
1358:
1355:
1348:
1347:
1344:Semiperimeter
1324:
1321:
1318:
1311:
1307:
1303:
1300:
1297:
1294:
1291:
1285:
1280:
1277:
1272:
1267:
1264:
1259:
1254:
1251:
1242:
1228:
1225:
1222:
1219:
1216:
1209:
1208:
1202:
1199:
1195:
1191:
1175:
1155:
1135:
1115:
1095:
1073:
1069:
1046:
1042:
1019:
1015:
1007:
991:
984:
968:
961:
960:semiperimeter
945:
925:
905:
885:
882:
879:
865:
846:
843:
838:
835:
827:
811:
807:
804:
798:
795:
787:
771:
768:
758:
754:
737:
731:
727:
721:
718:
710:
709:
708:
689:
685:
679:
673:
668:
662:
659:
651:
650:
649:
647:
643:
624:
619:
615:
609:
606:
598:
594:
591:
575:
572:
567:
564:
544:
539:
535:
529:
526:
518:
514:
497:
493:
488:
485:
477:
473:
457:
454:
451:
448:
440:
426:
421:
417:
411:
407:
401:
398:
390:
389:
388:
386:
370:
344:
340:
336:
331:
327:
323:
318:
314:
293:
290:
287:
284:
281:
261:
258:
255:
252:
249:
240:
231:
229:
225:
221:
217:
213:
209:
205:
201:
197:
187:
184:
180:
176:
159:
155:
148:
144:
133:
131:
127:
124:
118:
116:
112:
92:
90:
86:
82:
80:
76:
72:
70:
66:
62:
59:
56:
52:
48:
43:
38:
33:
19:
8901:>20 sides
8836:Decagon (10)
8821:Heptagon (7)
8811:Pentagon (5)
8801:Triangle (3)
8696:Equidiagonal
8632:
8547:
8516:
8507:
8499:
8490:
8481:
8461:10-orthoplex
8197:Dodecahedron
8143:
8118:
8107:
8096:
8087:
8078:
8069:
8065:
8055:
8047:
8043:
8035:
8031:
7970:
7928:
7924:
7918:
7898:
7891:
7866:
7858:
7834:
7827:
7796:
7785:
7774:. Retrieved
7770:
7760:
7749:. Retrieved
7744:
7734:
7689:
7677:
7629:
7567:
7561:
7541:
7532:
7528:
7515:
7482:
7478:
7472:
7463:
7459:
7446:
7437:
7433:
7420:
7412:
7407:
7398:
7394:
7381:
7372:
7368:
7355:
7346:
7340:
7330:
7325:
7317:
7312:
7301:
7280:
7243:
7239:
7214:
7201:
7152:
7140:
7091:
7061:
6981:
6926:
6920:
6914:
6881:
6877:
6864:
6823:
6819:
6806:
6787:Ternary plot
6742:Gateway Arch
6735:
6677:
6672:
6607:
6564:
6377:
6306:
6159:
5724:
5460:
5209:
4834:
4541:
4315:
4236:
4230:that can be
4163:
4024:
3964:
3850:trigonometry
3847:
3739:
3585:
3468:
3395:
3377:
3373:
3366:
3343:
3339:Fermat prime
3332:
3289:satisfy the
3149:circumcircle
3097:
2930:
2915:
2908:
2898:
2658:parallel to
2293:
2270:
2241:circumcenter
2222:
2187:
1190:circumcircle
1088:(tangent to
871:
863:
711:The area is
706:
646:trigonometry
641:
639:
391:The area is
362:
227:
199:
193:
9097:Star-shaped
9072:Rectilinear
9042:Equilateral
9037:Equiangular
9001:Hendecagram
8845:11–20 sides
8826:Octagon (8)
8816:Hexagon (6)
8791:Monogon (1)
8633:Equilateral
8470:10-demicube
8431:9-orthoplex
8381:8-orthoplex
8331:7-orthoplex
8288:6-orthoplex
8258:5-orthoplex
8213:Pentachoron
8201:Icosahedron
8176:Tetrahedron
7466:(1): 32–35.
6933:: 247–248.
6626:icosahedron
6618:tetrahedron
3220:. That is,
2256:Nagel point
2249:orthocenter
1883:Weitzenböck
872:A triangle
212:equiangular
9122:Categories
9102:Tangential
9006:Dodecagram
8784:1–10 sides
8775:By number
8756:Tangential
8736:Right kite
8456:10-simplex
8440:9-demicube
8390:8-demicube
8340:7-demicube
8297:6-demicube
8267:5-demicube
8181:Octahedron
7819:0031.06502
7776:2023-03-09
7751:2023-03-09
7726:0888.52012
7669:1396.51001
7608:0385.51006
7535:: 105–114.
7375:: 197–209.
7098:Birkhäuser
7037:1163.00008
6963:0116.12902
6906:1162.51305
6856:1163.26316
6798:References
6764:yield sign
6760:road signs
6622:octahedron
4166:reflection
3572:of 60° is
2586:are shown.
2233:Euler line
2210:The three
2203:The three
2196:The three
9082:Reinhardt
8991:Enneagram
8981:Heptagram
8971:Pentagram
8938:65537-gon
8796:Digon (2)
8766:Trapezoid
8731:Rectangle
8681:Bicentric
8643:Isosceles
8620:Triangles
8504:orthoplex
8426:9-simplex
8376:8-simplex
8326:7-simplex
8283:6-simplex
8253:5-simplex
8222:Tesseract
7972:MathWorld
7945:109362577
7691:Polyhedra
7661:125948074
7600:123776612
7246:: 97–109.
7193:118179744
7185:501976971
7132:118951675
7124:871539199
7070:124244932
7029:117769827
7021:775429168
6955:124726536
6898:123965364
6890:1443-5756
6848:115305257
6832:1443-5756
6700:simplexes
6610:polyhedra
6521:ω
6504:ω
6471:ω
6351:≤
6324:≤
6174:π
6082:≠
6045:−
6020:−
4454:∠
4425:∠
4396:∠
4232:inscribed
4219:altitudes
4086:×
4006:∘
3998:
3945:
3568:, as the
3376:Euclid's
2926:perimeter
2493:≥
2251:coincide.
2198:altitudes
1760:
1744:
1728:
1686:
1672:
1658:
1632:∘
1409:(Blundon)
1386:−
1301:−
294:γ
288:β
282:α
220:congruent
218:are also
9057:Isotoxal
9052:Isogonal
8996:Decagram
8986:Octagram
8976:Hexagram
8777:of sides
8706:Harmonic
8607:Polygons
8558:Topics:
8521:demicube
8486:polytope
8480:Uniform
8241:600-cell
8237:120-cell
8190:Demicube
8164:Pentagon
8144:Triangle
7793:(1948).
7718:41212721
7627:(2018).
7507:15049234
7499:27642581
7440:: 58–65.
7149:(2010).
6771:See also
6751:and the
6642:parallel
6593:are all
5768:divides
5745:on side
3697:so that
3378:Elements
2245:incenter
2229:centroid
1194:incircle
597:altitude
204:triangle
196:geometry
69:vertices
9077:Regular
9022:Concave
9015:Classes
8923:257-gon
8746:Rhombus
8686:Crossed
8495:simplex
8465:10-cube
8232:24-cell
8218:16-cell
8159:Hexagon
8013:regular
7811:4766401
7710:1458063
7592:1567647
7584:2689529
7013:2498836
6947:2687913
6840:2491926
6702:, with
6587:3.4.3.4
6579:3.4.6.4
5923:, then
4176:is the
4056:. Thus
3575:√
3561:√
2273:medians
2205:medians
2190:cevians
1006:exradii
757:apothem
183:degrees
9087:Simple
9032:Cyclic
9027:Convex
8751:Square
8691:Cyclic
8653:Obtuse
8648:Kepler
8435:9-cube
8385:8-cube
8335:7-cube
8292:6-cube
8262:5-cube
8149:Square
8026:Family
7943:
7906:
7879:
7846:
7817:
7809:
7771:Stella
7724:
7716:
7708:
7698:
7667:
7659:
7649:
7606:
7598:
7590:
7582:
7505:
7497:
7401:: 7–8.
7191:
7183:
7173:
7130:
7122:
7112:
7068:
7035:
7027:
7019:
7011:
7001:
6961:
6953:
6945:
6904:
6896:
6888:
6854:
6846:
6838:
6830:
6589:, and
6436:, and
6303:, then
5604:, and
5544:, and
5270:, and
5147:where
4895:, and
4662:, and
4602:, and
4515:, and
4446:, and
4228:square
3848:Using
3266:, and
3197:, and
2994:, and
2635:, and
2589:Lines
2416:, and
2356:, and
2296:point
2223:Every
1597:Angles
648:that:
387:that:
216:angles
9062:Magic
8658:Right
8638:Ideal
8628:Acute
8154:p-gon
7941:S2CID
7657:S2CID
7596:S2CID
7580:JSTOR
7558:(PDF)
7525:(PDF)
7503:S2CID
7495:JSTOR
7456:(PDF)
7430:(PDF)
7391:(PDF)
7365:(PDF)
7264:(PDF)
7236:(PDF)
7211:(PDF)
7189:S2CID
7128:S2CID
7066:S2CID
7058:(PDF)
7025:S2CID
6951:S2CID
6943:JSTOR
6929:(4).
6894:S2CID
6874:(PDF)
6844:S2CID
6816:(PDF)
6583:(3.6)
6567:tiles
5837:with
5564:from
2294:every
1205:Sides
759:, is
202:is a
198:, an
65:Edges
9092:Skew
8716:Kite
8611:List
8512:cube
8185:Cube
8015:and
7904:ISBN
7877:ISBN
7844:ISBN
7807:OCLC
7714:OCLC
7696:ISBN
7647:ISBN
7181:OCLC
7171:ISBN
7120:OCLC
7110:ISBN
7017:OCLC
6999:ISBN
6886:ISSN
6828:ISSN
6624:and
6575:3.12
6276:and
6097:and
5880:and
5814:and
4168:and
3872:and
3570:sine
2922:area
2779:and
2704:and
1799:Area
1192:and
1168:and
983:area
595:The
130:Area
67:and
54:Type
8061:(p)
7933:doi
7815:Zbl
7722:Zbl
7665:Zbl
7639:doi
7604:Zbl
7572:doi
7487:doi
7483:115
7163:doi
7102:doi
7033:Zbl
6991:doi
6959:Zbl
6935:doi
6902:Zbl
6852:Zbl
6591:3.6
6071:if
4388:of
4237:By
3995:sin
3942:sin
2811:As
1757:sin
1741:sin
1725:sin
1683:cos
1669:cos
1655:cos
557:or
519:is
478:is
194:In
188:60°
83:{3}
9124::
8566:•
8562:•
8542:21
8538:•
8535:k1
8531:•
8528:k2
8506:•
8463:•
8433:•
8411:21
8407:•
8404:41
8400:•
8397:42
8383:•
8361:21
8357:•
8354:31
8350:•
8347:32
8333:•
8311:21
8307:•
8304:22
8290:•
8260:•
8239:•
8220:•
8199:•
8183:•
8115:/
8104:/
8094:/
8085:/
8063:/
7969:.
7939:.
7929:24
7927:.
7875:.
7842:.
7813:.
7769:.
7743:.
7720:.
7712:.
7706:MR
7704:.
7688:.
7663:.
7655:.
7645:.
7616:^
7602:.
7594:.
7588:MR
7586:.
7578:.
7568:50
7566:.
7560:.
7549:;
7533:15
7531:.
7527:.
7501:.
7493:.
7481:.
7464:41
7462:.
7458:.
7438:10
7436:.
7432:.
7397:.
7393:.
7373:12
7371:.
7367:.
7288:^
7272:^
7252:^
7242:.
7238:.
7223:^
7213:.
7187:.
7179:.
7169:.
7126:.
7118:.
7108:.
7078:^
7060:.
7045:^
7031:.
7023:.
7015:.
7009:MR
7007:.
6997:.
6971:^
6957:.
6949:.
6941:.
6927:36
6925:.
6900:.
6892:.
6880:.
6876:.
6850:.
6842:.
6836:MR
6834:.
6824:10
6822:.
6818:.
6728:.
6620:,
6585:,
6581:,
6577:,
6573:.
6543:0.
6409:,
6219:12
6157:.
5584:,
5524:,
5434:11
5382:16
5250:,
4875:,
4642:,
4582:,
4495:,
4417:,
4302:2.
4180:,
4002:60
3580:/2
3381:.
3371:.
3317:.
3243:,
3174:,
3095:.
3034:,
2974:,
2753:,
2681:,
2612:,
2396:,
2336:,
2243:,
1628:60
1448:12
1292:25
1128:,
1108:,
1061:,
1034:,
1004:,
981:,
958:,
938:,
918:,
359:).
230:.
8613:)
8609:(
8599:e
8592:t
8585:v
8550:-
8548:n
8540:k
8533:2
8526:1
8519:-
8517:n
8510:-
8508:n
8502:-
8500:n
8493:-
8491:n
8484:-
8482:n
8409:4
8402:2
8395:1
8359:3
8352:2
8345:1
8309:2
8302:1
8131:n
8129:H
8122:2
8119:G
8111:4
8108:F
8100:8
8097:E
8091:7
8088:E
8082:6
8079:E
8070:n
8066:D
8059:2
8056:I
8048:n
8044:B
8036:n
8032:A
8004:e
7997:t
7990:v
7975:.
7947:.
7935::
7912:.
7885:.
7873:3
7852:.
7821:.
7779:.
7754:.
7728:.
7671:.
7641::
7610:.
7574::
7509:.
7489::
7399:1
7266:.
7244:4
7195:.
7165::
7134:.
7104::
7072:.
7039:.
6993::
6965:.
6937::
6908:.
6882:9
6858:.
6766:.
6755:.
6744:.
6716:2
6713:=
6710:n
6698:-
6686:n
6655:n
6652:2
6540:=
6535:3
6531:z
6525:2
6517:+
6512:2
6508:z
6501:+
6496:1
6492:z
6449:3
6445:z
6422:2
6418:z
6395:1
6391:z
6364:.
6359:7
6356:9
6344:2
6340:A
6334:1
6330:A
6319:9
6316:7
6289:2
6285:A
6262:1
6258:A
6232:,
6224:3
6215:1
6183:3
6178:3
6141:,
6136:y
6133:1
6128:=
6123:t
6120:1
6115:+
6110:q
6107:1
6085:q
6079:t
6053:2
6049:q
6040:2
6036:t
6028:3
6024:q
6015:3
6011:t
5986:,
5980:q
5977:+
5974:t
5967:2
5963:q
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5956:q
5953:t
5950:+
5945:2
5941:t
5934:=
5931:z
5911:y
5891:D
5888:P
5868:z
5848:A
5845:D
5825:A
5822:D
5802:D
5799:P
5779:A
5776:P
5756:C
5753:B
5733:D
5711:.
5706:2
5702:a
5698:=
5693:2
5689:t
5685:+
5682:t
5679:q
5676:+
5671:2
5667:q
5647:,
5644:t
5641:+
5638:q
5635:=
5632:p
5612:C
5592:B
5572:A
5552:t
5532:q
5512:p
5492:C
5489:B
5469:P
5447:.
5442:4
5438:a
5431:=
5427:)
5421:4
5417:t
5413:+
5408:4
5404:q
5400:+
5395:4
5391:p
5386:(
5363:,
5358:2
5354:a
5350:5
5347:=
5343:)
5337:2
5333:t
5329:+
5324:2
5320:q
5316:+
5311:2
5307:p
5302:(
5298:4
5278:t
5258:q
5238:p
5218:P
5195:P
5175:L
5155:R
5135:,
5131:]
5125:2
5121:L
5115:2
5111:R
5107:2
5104:+
5099:2
5094:)
5088:2
5084:L
5080:+
5075:2
5071:R
5066:(
5060:[
5056:3
5053:=
5048:4
5044:t
5040:+
5035:4
5031:q
5027:+
5022:4
5018:p
4998:,
4994:)
4988:2
4984:L
4980:+
4975:2
4971:R
4966:(
4962:3
4959:=
4954:2
4950:t
4946:+
4941:2
4937:q
4933:+
4928:2
4924:p
4903:t
4883:q
4863:p
4843:P
4821:.
4816:2
4811:)
4805:2
4801:a
4797:+
4792:2
4788:t
4784:+
4779:2
4775:q
4771:+
4766:2
4762:p
4757:(
4752:=
4748:)
4742:4
4738:a
4734:+
4729:4
4725:t
4721:+
4716:4
4712:q
4708:+
4703:4
4699:p
4694:(
4690:3
4670:C
4650:B
4630:A
4610:t
4590:q
4570:p
4550:P
4523:C
4503:B
4483:A
4463:A
4460:P
4457:C
4434:C
4431:P
4428:B
4405:B
4402:P
4399:A
4372:P
4344:P
4324:P
4299:=
4294:r
4291:R
4269:r
4249:R
4195:3
4190:D
4142:2
4138:a
4132:4
4128:3
4122:=
4119:b
4116:a
4111:4
4107:3
4101:=
4096:2
4092:3
4083:b
4080:a
4075:2
4072:1
4067:=
4064:A
4041:2
4037:3
4011:.
3992:b
3989:a
3984:2
3981:1
3976:=
3973:A
3951:.
3948:C
3939:b
3936:a
3931:2
3928:1
3923:=
3920:A
3900:C
3880:b
3860:a
3829:.
3824:2
3820:a
3814:4
3810:3
3804:=
3801:A
3781:h
3778:a
3773:2
3770:1
3748:h
3726:.
3723:a
3718:2
3714:3
3708:=
3705:h
3683:2
3679:a
3675:=
3670:2
3666:h
3662:+
3657:2
3652:)
3647:2
3644:a
3639:(
3614:a
3594:a
3582:.
3577:3
3563:3
3539:.
3536:h
3533:a
3528:2
3525:1
3520:=
3517:A
3497:h
3477:a
3449:a
3427:2
3423:a
3417:4
3413:3
3407:=
3404:A
3352:r
3301:P
3277:C
3274:P
3254:B
3251:P
3231:A
3228:P
3208:C
3205:P
3185:B
3182:P
3162:A
3159:P
3135:C
3132:B
3129:A
3109:P
3083:P
3063:,
3060:h
3057:=
3054:f
3051:+
3048:e
3045:+
3042:d
3022:h
3002:f
2982:e
2962:d
2942:P
2892:.
2880:C
2877:B
2874:A
2854:E
2851:H
2848:P
2828:H
2825:C
2822:G
2819:P
2805:.
2793:G
2790:D
2787:P
2767:I
2764:F
2761:P
2741:E
2738:H
2735:P
2715:A
2712:C
2692:C
2689:B
2669:B
2666:A
2646:I
2643:H
2623:G
2620:F
2600:E
2597:D
2574:C
2571:B
2568:A
2532:.
2527:2
2523:z
2519:+
2514:2
2510:y
2506:+
2501:2
2497:x
2489:)
2483:2
2479:r
2475:+
2470:2
2466:q
2462:+
2457:2
2453:p
2448:(
2444:4
2424:z
2404:y
2384:x
2364:r
2344:q
2324:p
2304:P
2262:.
2166:c
2162:r
2158:=
2153:b
2149:r
2145:=
2140:a
2136:r
2112:9
2106:c
2102:r
2098:+
2093:b
2089:r
2085:+
2080:a
2076:r
2069:=
2066:r
2043:2
2039:c
2035:+
2030:2
2026:b
2022:+
2017:2
2013:a
2009:=
2004:2
2000:R
1996:9
1974:r
1971:2
1968:=
1965:R
1934:3
1931:2
1926:)
1922:c
1919:b
1916:a
1913:(
1908:4
1904:3
1898:=
1895:T
1885:)
1881:(
1863:3
1858:4
1851:2
1847:c
1843:+
1838:2
1834:b
1830:+
1825:2
1821:a
1814:=
1811:T
1781:8
1778:1
1773:=
1768:2
1765:C
1752:2
1749:B
1736:2
1733:A
1702:2
1699:3
1694:=
1690:C
1680:+
1676:B
1666:+
1662:A
1624:=
1621:C
1618:=
1615:B
1612:=
1609:A
1581:R
1576:2
1570:3
1565:3
1559:=
1556:s
1535:r
1530:3
1525:3
1522:=
1519:s
1498:T
1493:3
1488:3
1485:=
1480:2
1476:s
1454:r
1451:R
1445:+
1440:2
1436:r
1432:3
1429:=
1424:2
1420:s
1397:r
1393:)
1389:4
1381:3
1376:3
1372:(
1368:+
1365:R
1362:2
1359:=
1356:s
1325:r
1322:R
1319:4
1312:2
1308:r
1304:2
1298:r
1295:R
1286:=
1281:c
1278:1
1273:+
1268:b
1265:1
1260:+
1255:a
1252:1
1229:c
1226:=
1223:b
1220:=
1217:a
1176:r
1156:R
1136:c
1116:b
1096:a
1074:c
1070:r
1047:b
1043:r
1020:a
1016:r
992:T
969:s
946:c
926:b
906:a
886:C
883:B
880:A
847:3
844:h
839:=
836:r
812:3
808:h
805:2
799:=
796:R
772:3
769:h
738:3
732:2
728:h
722:=
719:A
690:2
686:R
680:4
674:3
669:3
663:=
660:A
642:R
625:a
620:2
616:3
610:=
607:h
576:2
573:R
568:=
565:r
545:a
540:6
536:3
530:=
527:r
498:3
494:a
489:=
486:R
458:a
455:3
452:=
449:p
427:,
422:2
418:a
412:4
408:3
402:=
399:A
371:a
345:c
341:h
337:=
332:b
328:h
324:=
319:a
315:h
291:=
285:=
262:c
259:=
256:b
253:=
250:a
185:)
181:(
160:2
156:a
149:4
145:3
122:3
120:D
73:3
34:.
20:)
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