1062:
1902:
1582:
35:
549:
1255:
20:
1577:{\displaystyle {\begin{cases}{\overline {MA_{1}}}+{\overline {MA_{3}}}+\cdots +{\overline {MA_{n-2}}}+{\overline {MA_{n}}}<n/{\sqrt {2}}&{\text{if }}n{\text{ is odd}};\\{\overline {MA_{1}}}+{\overline {MA_{3}}}+\cdots +{\overline {MA_{n-3}}}+{\overline {MA_{n-1}}}\leq n/{\sqrt {2}}&{\text{if }}n{\text{ is even}}.\end{cases}}}
1891:
2166:
The reverse is true for all cyclic polygons with any number of sides; if all such central angles have rational tangents for their quarter angles then the implied cyclic polygon circumscribed by the unit circle will simultaneously have rational side lengths and rational area. Additionally, each
1150:
case, in each case replacing a side with three more sides and noting that these three new sides together with the old side form a quadrilateral which itself has this property; the alternate angles of the latter quadrilateral represent the additions to the alternate angle sums of the previous
1647:
2005:
1108:
For a cyclic polygon with an odd number of sides, all angles are equal if and only if the polygon is regular. A cyclic polygon with an even number of sides has all angles equal if and only if the alternate sides are equal (that is, sides
832:
2336:. These rational numbers are the tangents of the individual quarter angles, using the formula for the tangent of the difference of angles. Rational side lengths for the polygon circumscribed by the unit circle are thus obtained as
391:
1913:
is cyclic. Consider a unit circle, then circumscribe a regular triangle such that each side touches the circle. Circumscribe a circle, then circumscribe a square. Again circumscribe a circle, then circumscribe a regular
209:. (Because of this, some authors define "concyclic" only in the context of four or more points on a circle.) Several other sets of points defined from a triangle are also concyclic, with different circles; see
224:
of the circle on which lie a set of points is, by definition, the radius of the circumcircle of any triangle with vertices at any three of those points. If the pairwise distances among three of the points are
2167:
diagonal that connects two vertices, whether or not the two vertices are adjacent, will have a rational length. Such a cyclic polygon can be scaled so that its area and lengths are all integers.
1025:
936:
1886:{\displaystyle 3({\overline {MA_{1}}}^{2}+{\overline {MA_{2}}}^{2}+\dots +{\overline {MA_{n}}}^{2})^{2}=2n({\overline {MA_{1}}}^{4}+{\overline {MA_{2}}}^{4}+\dots +{\overline {MA_{n}}}^{4}).}
2163:. Every known Robbins pentagon (has diagonals that have rational length and) has this property, though it is an unsolved problem whether every possible Robbins pentagon has this property.
396:
The equation of the circumcircle of a triangle, and expressions for the radius and the coordinates of the circle's center, in terms of the
Cartesian coordinates of the vertices are given
613:
1628:
2434:, which is the smallest circle that completely contains a set of points. Every set of points in the plane has a unique minimum bounding circle, which may be constructed by a
1927:
1143:, the sum of one set of alternate angles (the first, third, fifth, etc.) equals the sum of the other set of alternate angles. This can be proven by induction from the
2048:(geometric transformations generated by reflections and circle inversions), as these transformations preserve the concyclicity of points only in this extended sense.
668:
509:
485:
2918:
2074:
of the plane), concyclicity has a particularly simple formulation: four points in the complex plane are either concyclic or collinear if and only if their
243:
1132:. In all known cases, its diagonals also have rational lengths, though whether this is true for all possible Robbins pentagons is an unsolved problem.
2090:
Some cyclic polygons have the property that their area and all of their side lengths are positive integers. Triangles with this property are called
456:
to the sides of a triangle, then the six points of intersection of the lines and the sides of the triangle are concyclic, in what is called the
2441:
Even if a set of points are concyclic, their circumscribing circle may be different from their minimum bounding circle. For example, for an
2118:
be the angle spanned by one side of the cyclic polygon as viewed from the center of the circumscribing circle. Similarly define the
2017:
2094:; cyclic quadrilaterals with this property (and that the diagonals that connect opposite vertices have integer length) are called
2170:
This reverse relationship gives a way to generate cyclic polygons with integer area, sides, and diagonals. For a polygon with
2146:
sides. Every
Heronian triangle and every Brahmagupta quadrilateral has a rational value for the tangent of the quarter angle,
864:
in order, then it is cyclic if and only if the product of the diagonals equals the sum of the products of opposite sides:
975:
2040:(points along a single line) are considered to be concyclic. This point of view is helpful, for instance, when studying
3083:
2682:
870:
3000:
2973:
2946:
2720:
2597:
2521:
2473:
2878:
124:. However, four or more points in the plane are not necessarily concyclic. After triangles, the special case of
1166:
tangent to each side of the polygon; these tangency points are thus concyclic on the inscribed circle. Let one
2445:, the minimum bounding circle has the longest side as diameter and does not pass through the opposite vertex.
3099:
2021:
574:
1594:
3104:
2000:{\displaystyle \prod _{n=3}^{\infty }{\frac {1}{\cos \left({\frac {\pi }{n}}\right)}}=8.7000366\ldots .}
2736:
De
Villiers, Michael (March 2011). "95.14 Equiangular cyclic and equilateral circumscribed polygons".
2095:
1061:
1264:
2787:
2738:
2056:
1046:
2045:
397:
2431:
1045:(has perpendicular diagonals) if and only if the midpoints of the sides and the feet of the four
616:
2516:, Pure and Applied Undergraduate Texts, vol. 8, American Mathematical Society, p. 63,
161:
2990:
2490:
827:{\displaystyle R={\frac {1}{4}}{\sqrt {\frac {(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}}},}
2963:
2936:
2912:
2710:
2587:
2511:
2463:
2071:
1066:
2810:
2572:
2557:
845:
565:
553:
543:
183:
bisectors, and the concyclic condition is that they all meet in a single point, the centre
125:
8:
3078:
620:
453:
423:
413:
214:
3037:
2890:
2763:
2755:
2656:
2538:
2033:
1159:
494:
470:
446:
2826:
Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle
2614:
3060:
2996:
2969:
2942:
2767:
2716:
2688:
2678:
2593:
2517:
2469:
2418:
is concyclic. This property can be thought of as an analogue for concyclicity of the
2406:. These can be made into integers by scaling the side lengths by a shared constant.
2195:
be rational numbers. These are the tangents of one quarter of the cumulative angles
2091:
512:
464:
435:
409:
408:
In any triangle all of the following nine points are concyclic on what is called the
210:
173:
86:
62:
3041:
3029:
2932:
2900:
2829:
2796:
2747:
2706:
2648:
2570:
Scott, J. A. "Some examples of the use of areal coordinates in triangle geometry",
2099:
2041:
2037:
1163:
1129:
1098:
1035:
530:
are concyclic, with the segment from the circumcenter to the
Lemoine point being a
427:
386:{\displaystyle R={\sqrt {\frac {a^{2}b^{2}c^{2}}{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}}.}
66:
23:
2852:
1097:. A polygon is cyclic if and only if the perpendicular bisectors of its edges are
848:, if a quadrilateral is given by the pairwise distances between its four vertices
2806:
2553:
2442:
2103:
1910:
1125:
1102:
109:
619:) which is true if and only if the opposite angles inside the quadrilateral are
3063:
2833:
2419:
2060:
569:
527:
457:
2801:
2782:
2751:
1918:, and so on. The radii of the circumscribed circles converge to the so-called
3093:
2692:
2507:
2119:
2052:
1042:
640:
523:
450:
113:
2904:
1070:
838:
519:
488:
439:
431:
206:
196:
121:
2414:
A set of five or more points is concyclic if and only if every four-point
1901:
2592:, MAA Spectrum (2nd ed.), Cambridge University Press, p. xxii,
2435:
2079:
2075:
417:
102:
2759:
548:
34:
2968:, MAA Spectrum (2nd ed.), Cambridge University Press, p. 65,
2660:
3016:
Megiddo, N. (1983). "Linear-time algorithms for linear programming in
3068:
2675:
The
Algebra of Geometry: Cartesian, Areal and Projective Co-Ordinates
2639:
Hoehn, Larry (March 2000), "Circumradius of a cyclic quadrilateral",
837:
an expression that was derived by the Indian mathematician
Vatasseri
27:
3033:
2652:
1034:
may be internal or external to the circle. This theorem is known as
2895:
1915:
1121:
1074:
531:
202:
117:
58:
2102:. More generally, versions of these cyclic polygons scaled by a
1090:
82:
2539:"The circles of Lester, Evans, Parry, and their generalizations"
2415:
1182:
on the circle, the product of the perpendicular distances from
1174:-gon be tangential to that circle at the vertices of the first
221:
78:
2938:
The
Advanced Geometry of Plane Curves and Their Applications
2465:
1190:-gon equals the product of the perpendicular distances from
19:
2106:
will have area and side lengths that are rational numbers.
2012:
1570:
412:: the midpoints of the three edges, the feet of the three
2879:"Cyclic Averages of Regular Polygons and Platonic Solids"
1930:
1650:
1597:
1258:
979:
978:
873:
671:
577:
497:
473:
246:
2704:
1020:{\displaystyle \displaystyle AX\cdot XC=BX\cdot XD.}
2715:, Mathematical Association of America, p. 77,
3058:
2780:
2032:In contexts where lines are taken to be a type of
1999:
1896:
1885:
1622:
1576:
1049:are eight concyclic points, on what is called the
1019:
930:
826:
607:
503:
479:
385:
2917:: CS1 maint: DOI inactive as of September 2024 (
2781:Buchholz, Ralph H.; MacDougall, James A. (2008).
2098:; cyclic pentagons with this property are called
1905:A sequence of circumscribed polygons and circles.
931:{\displaystyle AC\cdot BD=AB\cdot CD+BC\cdot AD.}
3091:
2085:
1093:in which all vertices are concyclic is called a
662:) / 2 has its circumradius given by
1170:-gon be inscribed in a circle, and let another
623:. A cyclic quadrilateral with successive sides
2883:Communications in Mathematics and Applications
2783:"Cyclic polygons with rational sides and area"
2468:, Jones & Bartlett Learning, p. 21,
491:of the six triangles that are defined inside
2876:
2615:"On the diagonals of a cyclic quadrilateral"
2612:
1201:
1069:chord formula, the area bounded by the
2995:, Courier Dover Publications, p. 431,
2735:
2941:, Courier Dover Publications, p. 24,
2495:, Swan Sonnenschein & co., p. 126
2425:
2020:). The reciprocal of this constant is the
131:
2894:
2800:
2613:Alsina, Claudi; Nelsen, Roger B. (2007),
2461:
403:
120:is a cyclic polygon, with a well-defined
1900:
1060:
547:
537:
33:
18:
3015:
2931:
2823:
2672:
2488:
1226:on the unit circle. Then for any point
3092:
2870:
2606:
2506:
941:If two lines, one containing segment
3059:
2988:
2838:Republished by Dover Publications as
2638:
2585:
608:{\displaystyle \angle CAD=\angle CBD}
416:, and the points halfway between the
2961:
1634:on the circumcircle to the vertices
1623:{\displaystyle {\overline {MA_{i}}}}
564:with concyclic vertices is called a
2828:. Houghton Mifflin Co. p. 72.
2536:
2409:
467:associated with any given triangle
128:has been most extensively studied.
13:
3084:The Wolfram Demonstrations Project
1947:
1056:
593:
578:
14:
3116:
3052:
2430:A related notion is the one of a
1630:are the distances from any point
945:and the other containing segment
2992:Geometry: A Comprehensive Course
2673:Bradley, Christopher J. (2007),
552:Four concyclic points forming a
420:and each of the three vertices.
3009:
2982:
2955:
2925:
2845:
2817:
2774:
2729:
2698:
2677:, Highperception, p. 179,
1920:polygon circumscribing constant
1897:Polygon circumscribing constant
156:are equal distances. Therefore
105:from the center of the circle.
93:, and the circle is called its
2712:Methods for Euclidean Geometry
2666:
2632:
2579:
2564:
2530:
2500:
2482:
2455:
1877:
1775:
1757:
1654:
814:
802:
799:
787:
784:
772:
769:
757:
752:
734:
731:
713:
710:
692:
373:
355:
352:
334:
331:
310:
307:
289:
237:, then the circle's radius is
1:
2877:Meskhishvili, Mamuka (2020).
2705:Byer, Owen; Lazebnik, Felix;
2513:Geometry for College Students
2448:
2086:Integer area and side lengths
2027:
1128:sides and area is known as a
969:are concyclic if and only if
2965:Complex Numbers and Geometry
2589:Circles: A Mathematical View
1866:
1828:
1796:
1745:
1707:
1675:
1615:
1526:
1495:
1458:
1433:
1371:
1346:
1309:
1284:
205:fall on a circle called the
190:
140:of a circle on which points
7:
2840:Advanced Euclidean Geometry
2576:83, November 1999, 472–477.
1194:to the sides of the second
101:. All concyclic points are
26:perpendicular bisectors of
10:
3121:
2853:"Inequalities proposed in
2824:Johnson, Roger A. (1929).
2462:Libeskind, Shlomo (2008),
2096:Brahmagupta quadrilaterals
2042:inversion through a circle
1186:to the sides of the first
1178:-gon. Then from any point
1041:A convex quadrilateral is
556:, showing two equal angles
541:
194:
172:distinct points there are
112:that do not all fall on a
89:are concyclic is called a
77:) if they lie on a common
3022:SIAM Journal on Computing
2962:Hahn, Liang-shin (1996),
2802:10.1016/j.jnt.2007.05.005
2752:10.1017/S0025557200002461
1202:Point on the circumcircle
16:Points on a common circle
2788:Journal of Number Theory
2739:The Mathematical Gazette
2366:. The rational area is
2057:real and imaginary parts
2022:Kepler–Bouwkamp constant
1249:to the vertices satisfy
116:are concyclic, so every
30:between concyclic points
3020:and related problems".
2905:10.26713/cma.v11i3.1420
2432:minimum bounding circle
2426:Minimum bounding circle
2055:(formed by viewing the
953:, then the four points
617:inscribed angle theorem
132:Perpendicular bisectors
3082:by Michael Schreiber,
2907:(inactive 2024-09-12).
2489:Elliott, John (1902),
2046:Möbius transformations
2036:with infinite radius,
2001:
1951:
1906:
1887:
1624:
1578:
1086:
1077:of every unit regular
1065:As a corollary of the
1021:
932:
828:
609:
557:
505:
481:
449:are drawn through the
404:Other concyclic points
387:
201:The vertices of every
162:perpendicular bisector
148:lie must be such that
136:In general the centre
54:
38:Circumscribed circle,
31:
3079:Four Concyclic Points
2159:, for every value of
2072:Cartesian coordinates
2002:
1931:
1904:
1888:
1625:
1579:
1245:, the distances from
1113:are equal, and sides
1105:is a cyclic polygon.
1064:
1022:
933:
841:in the 15th century.
829:
610:
551:
538:Cyclic quadrilaterals
506:
482:
388:
126:cyclic quadrilaterals
95:circumscribing circle
37:
22:
2707:Smeltzer, Deirdre L.
2641:Mathematical Gazette
2573:Mathematical Gazette
1928:
1648:
1595:
1256:
976:
871:
669:
575:
566:cyclic quadrilateral
554:cyclic quadrilateral
544:Cyclic quadrilateral
526:, and its first two
495:
471:
244:
164:of the line segment
108:Three points in the
42:, and circumcenter,
3100:Elementary geometry
2989:Pedoe, Dan (1988),
2855:Crux Mathematicorum
2834:2027/wu.89043163211
2622:Forum Geometricorum
2586:Pedoe, Dan (1997),
2546:Forum Geometricorum
2492:Elementary Geometry
1210:-gon have vertices
426:states that in any
3105:Incidence geometry
3061:Weisstein, Eric W.
2866:. p. 190, #332.10.
2864:The IMO Compendium
2537:Yiu, Paul (2010),
2139:for the remaining
2092:Heronian triangles
2044:or more generally
2034:generalised circle
1997:
1907:
1883:
1620:
1574:
1569:
1160:tangential polygon
1089:More generally, a
1087:
1051:eight-point circle
1017:
1016:
928:
824:
605:
558:
501:
477:
383:
55:
32:
2508:Isaacs, I. Martin
2100:Robbins pentagons
1983:
1976:
1869:
1831:
1799:
1748:
1710:
1678:
1618:
1562:
1554:
1547:
1529:
1498:
1461:
1436:
1407:
1399:
1392:
1374:
1349:
1312:
1287:
1230:on the minor arc
1162:is one having an
1030:The intersection
846:Ptolemy's theorem
819:
818:
686:
504:{\displaystyle T}
480:{\displaystyle T}
465:van Lamoen circle
436:nine-point center
410:nine-point circle
378:
377:
211:Nine-point circle
181: − 1)/2
3112:
3074:
3073:
3046:
3045:
3013:
3007:
3005:
2986:
2980:
2978:
2959:
2953:
2951:
2929:
2923:
2922:
2916:
2908:
2898:
2874:
2868:
2867:
2861:
2849:
2843:
2842:, 1960 and 2007.
2837:
2821:
2815:
2814:
2804:
2778:
2772:
2771:
2746:(532): 102–107.
2733:
2727:
2725:
2702:
2696:
2695:
2670:
2664:
2663:
2636:
2630:
2629:
2619:
2610:
2604:
2602:
2583:
2577:
2568:
2562:
2560:
2543:
2534:
2528:
2526:
2504:
2498:
2496:
2486:
2480:
2478:
2459:
2422:of convex sets.
2410:Other properties
2405:
2365:
2335:
2324:
2275:
2254:
2238:
2219:
2203:
2194:
2173:
2162:
2158:
2145:
2138:
2117:
2038:collinear points
2015:
2006:
2004:
2003:
1998:
1984:
1982:
1981:
1977:
1969:
1953:
1950:
1945:
1892:
1890:
1889:
1884:
1876:
1875:
1870:
1865:
1864:
1863:
1850:
1838:
1837:
1832:
1827:
1826:
1825:
1812:
1806:
1805:
1800:
1795:
1794:
1793:
1780:
1765:
1764:
1755:
1754:
1749:
1744:
1743:
1742:
1729:
1717:
1716:
1711:
1706:
1705:
1704:
1691:
1685:
1684:
1679:
1674:
1673:
1672:
1659:
1640:
1633:
1629:
1627:
1626:
1621:
1619:
1614:
1613:
1612:
1599:
1590:
1583:
1581:
1580:
1575:
1573:
1572:
1563:
1560:
1555:
1552:
1548:
1543:
1541:
1530:
1525:
1524:
1523:
1504:
1499:
1494:
1493:
1492:
1473:
1462:
1457:
1456:
1455:
1442:
1437:
1432:
1431:
1430:
1417:
1408:
1405:
1400:
1397:
1393:
1388:
1386:
1375:
1370:
1369:
1368:
1355:
1350:
1345:
1344:
1343:
1324:
1313:
1308:
1307:
1306:
1293:
1288:
1283:
1282:
1281:
1268:
1248:
1244:
1229:
1225:
1209:
1197:
1193:
1189:
1185:
1181:
1177:
1173:
1169:
1164:inscribed circle
1154:
1149:
1142:
1138:
1130:Robbins pentagon
1116:
1112:
1084:
1080:
1036:power of a point
1026:
1024:
1023:
1018:
937:
935:
934:
929:
833:
831:
830:
825:
820:
817:
755:
690:
689:
687:
679:
614:
612:
611:
606:
560:A quadrilateral
510:
508:
507:
502:
486:
484:
483:
478:
428:scalene triangle
424:Lester's theorem
392:
390:
389:
384:
379:
376:
287:
286:
285:
276:
275:
266:
265:
255:
254:
215:Lester's theorem
160:must lie on the
53:
45:
41:
3120:
3119:
3115:
3114:
3113:
3111:
3110:
3109:
3090:
3089:
3055:
3050:
3049:
3034:10.1137/0212052
3014:
3010:
3003:
2987:
2983:
2976:
2960:
2956:
2949:
2930:
2926:
2910:
2909:
2875:
2871:
2859:
2851:
2850:
2846:
2822:
2818:
2779:
2775:
2734:
2730:
2723:
2703:
2699:
2685:
2671:
2667:
2653:10.2307/3621477
2637:
2633:
2617:
2611:
2607:
2600:
2584:
2580:
2569:
2565:
2541:
2535:
2531:
2524:
2505:
2501:
2487:
2483:
2476:
2460:
2456:
2451:
2443:obtuse triangle
2428:
2412:
2403:
2394:
2385:
2376:
2367:
2363:
2354:
2345:
2337:
2326:
2322:
2313:
2304:
2294:
2285:
2277:
2274:
2264:
2256:
2253:
2246:
2240:
2237:
2227:
2221:
2218:
2211:
2205:
2202:
2196:
2192:
2182:
2175:
2171:
2160:
2156:
2147:
2140:
2137:
2128:
2122:
2116:
2110:
2104:rational number
2088:
2030:
2011:
1968:
1964:
1957:
1952:
1946:
1935:
1929:
1926:
1925:
1911:regular polygon
1899:
1871:
1859:
1855:
1851:
1849:
1848:
1833:
1821:
1817:
1813:
1811:
1810:
1801:
1789:
1785:
1781:
1779:
1778:
1760:
1756:
1750:
1738:
1734:
1730:
1728:
1727:
1712:
1700:
1696:
1692:
1690:
1689:
1680:
1668:
1664:
1660:
1658:
1657:
1649:
1646:
1645:
1639:
1635:
1631:
1608:
1604:
1600:
1598:
1596:
1593:
1592:
1588:
1568:
1567:
1559:
1551:
1549:
1542:
1537:
1513:
1509:
1505:
1503:
1482:
1478:
1474:
1472:
1451:
1447:
1443:
1441:
1426:
1422:
1418:
1416:
1413:
1412:
1404:
1396:
1394:
1387:
1382:
1364:
1360:
1356:
1354:
1333:
1329:
1325:
1323:
1302:
1298:
1294:
1292:
1277:
1273:
1269:
1267:
1260:
1259:
1257:
1254:
1253:
1246:
1242:
1237:
1231:
1227:
1223:
1217:
1211:
1207:
1204:
1195:
1191:
1187:
1183:
1179:
1175:
1171:
1167:
1152:
1144:
1140:
1139:-gon with even
1136:
1114:
1110:
1103:regular polygon
1082:
1078:
1059:
1057:Cyclic polygons
977:
974:
973:
949:, intersect at
872:
869:
868:
756:
691:
688:
678:
670:
667:
666:
576:
573:
572:
568:; this happens
546:
540:
496:
493:
492:
472:
469:
468:
442:are concyclic.
406:
288:
281:
277:
271:
267:
261:
257:
256:
253:
245:
242:
241:
199:
193:
134:
69:are said to be
51:
43:
39:
17:
12:
11:
5:
3118:
3108:
3107:
3102:
3088:
3087:
3075:
3054:
3053:External links
3051:
3048:
3047:
3028:(4): 759–776.
3008:
3001:
2981:
2974:
2954:
2947:
2924:
2869:
2844:
2816:
2773:
2728:
2721:
2697:
2684:978-1906338008
2683:
2665:
2647:(499): 69–70,
2631:
2605:
2598:
2578:
2563:
2529:
2522:
2499:
2481:
2474:
2453:
2452:
2450:
2447:
2427:
2424:
2420:Helly property
2411:
2408:
2399:
2390:
2381:
2372:
2359:
2350:
2341:
2317:
2309:
2299:
2290:
2281:
2269:
2260:
2251:
2244:
2232:
2225:
2216:
2209:
2200:
2187:
2183:< ... <
2180:
2152:
2133:
2126:
2120:central angles
2114:
2087:
2084:
2061:complex number
2029:
2026:
2008:
2007:
1996:
1993:
1990:
1987:
1980:
1975:
1972:
1967:
1963:
1960:
1956:
1949:
1944:
1941:
1938:
1934:
1898:
1895:
1894:
1893:
1882:
1879:
1874:
1868:
1862:
1858:
1854:
1847:
1844:
1841:
1836:
1830:
1824:
1820:
1816:
1809:
1804:
1798:
1792:
1788:
1784:
1777:
1774:
1771:
1768:
1763:
1759:
1753:
1747:
1741:
1737:
1733:
1726:
1723:
1720:
1715:
1709:
1703:
1699:
1695:
1688:
1683:
1677:
1671:
1667:
1663:
1656:
1653:
1637:
1617:
1611:
1607:
1603:
1587:For a regular
1585:
1584:
1571:
1566:
1558:
1550:
1546:
1540:
1536:
1533:
1528:
1522:
1519:
1516:
1512:
1508:
1502:
1497:
1491:
1488:
1485:
1481:
1477:
1471:
1468:
1465:
1460:
1454:
1450:
1446:
1440:
1435:
1429:
1425:
1421:
1415:
1414:
1411:
1403:
1395:
1391:
1385:
1381:
1378:
1373:
1367:
1363:
1359:
1353:
1348:
1342:
1339:
1336:
1332:
1328:
1322:
1319:
1316:
1311:
1305:
1301:
1297:
1291:
1286:
1280:
1276:
1272:
1266:
1265:
1263:
1240:
1235:
1221:
1215:
1203:
1200:
1135:In any cyclic
1095:cyclic polygon
1058:
1055:
1028:
1027:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
939:
938:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
835:
834:
823:
816:
813:
810:
807:
804:
801:
798:
795:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
762:
759:
754:
751:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
700:
697:
694:
685:
682:
677:
674:
604:
601:
598:
595:
592:
589:
586:
583:
580:
570:if and only if
542:Main article:
539:
536:
528:Brocard points
500:
476:
458:Lemoine circle
405:
402:
394:
393:
382:
375:
372:
369:
366:
363:
360:
357:
354:
351:
348:
345:
342:
339:
336:
333:
330:
327:
324:
321:
318:
315:
312:
309:
306:
303:
300:
297:
294:
291:
284:
280:
274:
270:
264:
260:
252:
249:
195:Main article:
192:
189:
133:
130:
91:cyclic polygon
48:cyclic polygon
15:
9:
6:
4:
3:
2:
3117:
3106:
3103:
3101:
3098:
3097:
3095:
3085:
3081:
3080:
3076:
3071:
3070:
3065:
3062:
3057:
3056:
3043:
3039:
3035:
3031:
3027:
3023:
3019:
3012:
3004:
3002:9780486658124
2998:
2994:
2993:
2985:
2977:
2975:9780883855102
2971:
2967:
2966:
2958:
2950:
2948:9780486442761
2944:
2940:
2939:
2934:
2928:
2920:
2914:
2906:
2902:
2897:
2892:
2888:
2884:
2880:
2873:
2865:
2858:
2856:
2848:
2841:
2835:
2831:
2827:
2820:
2812:
2808:
2803:
2798:
2794:
2790:
2789:
2784:
2777:
2769:
2765:
2761:
2757:
2753:
2749:
2745:
2741:
2740:
2732:
2724:
2722:9780883857632
2718:
2714:
2713:
2708:
2701:
2694:
2690:
2686:
2680:
2676:
2669:
2662:
2658:
2654:
2650:
2646:
2642:
2635:
2627:
2623:
2616:
2609:
2601:
2599:9780883855188
2595:
2591:
2590:
2582:
2575:
2574:
2567:
2559:
2555:
2551:
2547:
2540:
2533:
2525:
2523:9780821847947
2519:
2515:
2514:
2509:
2503:
2494:
2493:
2485:
2477:
2475:9780763743666
2471:
2467:
2466:
2458:
2454:
2446:
2444:
2439:
2437:
2433:
2423:
2421:
2417:
2407:
2402:
2398:
2393:
2389:
2384:
2380:
2375:
2370:
2362:
2358:
2353:
2349:
2344:
2340:
2333:
2329:
2320:
2316:
2312:
2308:
2302:
2298:
2293:
2289:
2284:
2280:
2272:
2268:
2263:
2259:
2250:
2243:
2235:
2231:
2224:
2215:
2208:
2199:
2190:
2186:
2179:
2168:
2164:
2155:
2151:
2143:
2136:
2132:
2125:
2121:
2113:
2107:
2105:
2101:
2097:
2093:
2083:
2081:
2077:
2073:
2070:
2066:
2062:
2058:
2054:
2053:complex plane
2049:
2047:
2043:
2039:
2035:
2025:
2023:
2019:
2014:
1994:
1991:
1988:
1985:
1978:
1973:
1970:
1965:
1961:
1958:
1954:
1942:
1939:
1936:
1932:
1924:
1923:
1922:
1921:
1917:
1912:
1903:
1880:
1872:
1860:
1856:
1852:
1845:
1842:
1839:
1834:
1822:
1818:
1814:
1807:
1802:
1790:
1786:
1782:
1772:
1769:
1766:
1761:
1751:
1739:
1735:
1731:
1724:
1721:
1718:
1713:
1701:
1697:
1693:
1686:
1681:
1669:
1665:
1661:
1651:
1644:
1643:
1642:
1609:
1605:
1601:
1564:
1561: is even
1556:
1544:
1538:
1534:
1531:
1520:
1517:
1514:
1510:
1506:
1500:
1489:
1486:
1483:
1479:
1475:
1469:
1466:
1463:
1452:
1448:
1444:
1438:
1427:
1423:
1419:
1409:
1401:
1389:
1383:
1379:
1376:
1365:
1361:
1357:
1351:
1340:
1337:
1334:
1330:
1326:
1320:
1317:
1314:
1303:
1299:
1295:
1289:
1278:
1274:
1270:
1261:
1252:
1251:
1250:
1243:
1234:
1224:
1214:
1206:Let a cyclic
1199:
1165:
1161:
1156:
1147:
1133:
1131:
1127:
1123:
1118:
1106:
1104:
1100:
1096:
1092:
1076:
1072:
1068:
1063:
1054:
1052:
1048:
1044:
1043:orthodiagonal
1039:
1037:
1033:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
972:
971:
970:
968:
964:
960:
956:
952:
948:
944:
925:
922:
919:
916:
913:
910:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
877:
874:
867:
866:
865:
863:
859:
855:
851:
847:
842:
840:
821:
811:
808:
805:
796:
793:
790:
781:
778:
775:
766:
763:
760:
749:
746:
743:
740:
737:
728:
725:
722:
719:
716:
707:
704:
701:
698:
695:
683:
680:
675:
672:
665:
664:
663:
661:
657:
653:
649:
645:
642:
641:semiperimeter
638:
634:
630:
626:
622:
621:supplementary
618:
602:
599:
596:
590:
587:
584:
581:
571:
567:
563:
555:
550:
545:
535:
533:
529:
525:
524:Lemoine point
521:
518:A triangle's
516:
514:
511:by its three
498:
490:
489:circumcenters
487:contains the
474:
466:
461:
459:
455:
452:
451:Lemoine point
448:
443:
441:
437:
433:
432:Fermat points
429:
425:
421:
419:
415:
411:
401:
399:
380:
370:
367:
364:
361:
358:
349:
346:
343:
340:
337:
328:
325:
322:
319:
316:
313:
304:
301:
298:
295:
292:
282:
278:
272:
268:
262:
258:
250:
247:
240:
239:
238:
236:
232:
228:
223:
218:
216:
212:
208:
204:
198:
188:
186:
182:
180:
176:
171:
167:
163:
159:
155:
151:
147:
143:
139:
129:
127:
123:
119:
115:
114:straight line
111:
106:
104:
100:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
49:
36:
29:
25:
21:
3077:
3067:
3025:
3021:
3017:
3011:
2991:
2984:
2964:
2957:
2937:
2927:
2913:cite journal
2886:
2882:
2872:
2863:
2854:
2847:
2839:
2825:
2819:
2795:(1): 17–48.
2792:
2786:
2776:
2743:
2737:
2731:
2711:
2700:
2674:
2668:
2644:
2640:
2634:
2625:
2621:
2608:
2588:
2581:
2571:
2566:
2549:
2545:
2532:
2512:
2502:
2491:
2484:
2464:
2457:
2440:
2429:
2413:
2400:
2396:
2391:
2387:
2382:
2378:
2373:
2368:
2360:
2356:
2351:
2347:
2342:
2338:
2331:
2327:
2318:
2314:
2310:
2306:
2300:
2296:
2291:
2287:
2282:
2278:
2270:
2266:
2261:
2257:
2248:
2241:
2233:
2229:
2222:
2213:
2206:
2197:
2188:
2184:
2177:
2169:
2165:
2153:
2149:
2141:
2134:
2130:
2123:
2111:
2108:
2089:
2068:
2064:
2050:
2031:
2009:
1919:
1908:
1586:
1406: is odd
1238:
1232:
1219:
1212:
1205:
1157:
1145:
1134:
1119:
1117:are equal).
1107:
1094:
1088:
1071:circumcircle
1050:
1040:
1031:
1029:
966:
962:
958:
954:
950:
946:
942:
940:
861:
857:
853:
849:
843:
839:Parameshvara
836:
659:
655:
651:
647:
643:
636:
632:
628:
624:
561:
559:
520:circumcenter
517:
462:
444:
440:circumcenter
422:
407:
395:
234:
230:
226:
219:
207:circumcircle
200:
197:Circumcircle
184:
178:
174:
169:
165:
157:
153:
149:
145:
141:
137:
135:
122:circumcircle
107:
99:circumcircle
98:
94:
90:
74:
70:
56:
47:
3064:"Concyclic"
2933:Zwikker, C.
2889:: 335–355.
2552:: 175–209,
2438:algorithm.
2436:linear time
2174:sides, let
2080:real number
2076:cross-ratio
418:orthocenter
103:equidistant
3094:Categories
2896:2010.12340
2449:References
2330:= 2, ...,
2276:, and let
2028:Variations
2010:(sequence
1115:2, 4, 6, …
1111:1, 3, 5, …
1099:concurrent
438:, and the
430:, the two
24:Concurrent
3069:MathWorld
2768:233361080
2693:213434422
2395:) / (1 +
2305:) / (1 +
1992:…
1989:8.7000366
1971:π
1962:
1948:∞
1933:∏
1867:¯
1843:⋯
1829:¯
1797:¯
1746:¯
1722:⋯
1708:¯
1676:¯
1616:¯
1591:-gon, if
1532:≤
1527:¯
1518:−
1496:¯
1487:−
1467:⋯
1459:¯
1434:¯
1372:¯
1347:¯
1338:−
1318:⋯
1310:¯
1285:¯
1120:A cyclic
1047:altitudes
1005:⋅
987:⋅
917:⋅
899:⋅
881:⋅
809:−
794:−
779:−
764:−
594:∠
579:∠
414:altitudes
368:−
341:−
314:−
191:Triangles
71:concyclic
3042:14467740
2935:(2005),
2760:23248632
2709:(2010),
2510:(2009),
2228:+ ... +
1916:pentagon
1641:, then
1553:if
1398:if
1126:rational
1122:pentagon
1101:. Every
1081:-gon is
1075:incircle
532:diameter
454:parallel
203:triangle
118:triangle
87:vertices
75:cocyclic
59:geometry
2811:2382768
2661:3621477
2628:: 147–9
2558:2868943
2355:/ (1 +
2239:. Let
2220:, ...,
2193:< +∞
2176:0 <
2129:, ...,
2063:as the
2051:In the
2016:in the
2013:A051762
1091:polygon
1067:annulus
513:medians
83:polygon
46:, of a
3040:
2999:
2972:
2945:
2809:
2766:
2758:
2719:
2691:
2681:
2659:
2596:
2556:
2520:
2472:
2416:subset
2265:= 1 /
2255:, let
1198:-gon.
1155:-gon.
860:, and
615:(the
522:, its
434:, the
233:, and
222:radius
168:. For
85:whose
79:circle
67:points
28:chords
3038:S2CID
2891:arXiv
2860:(PDF)
2764:S2CID
2756:JSTOR
2657:JSTOR
2618:(PDF)
2542:(PDF)
2386:(1 −
2078:is a
2059:of a
1218:, …,
1124:with
447:lines
110:plane
2997:ISBN
2970:ISBN
2943:ISBN
2919:link
2717:ISBN
2689:OCLC
2679:ISBN
2594:ISBN
2518:ISBN
2470:ISBN
2325:for
2148:tan
2109:Let
2067:and
2018:OEIS
1909:Any
1377:<
1073:and
639:and
562:ABCD
463:The
398:here
220:The
213:and
152:and
144:and
81:. A
73:(or
61:, a
3030:doi
2901:doi
2830:hdl
2797:doi
2793:128
2748:doi
2649:doi
2371:= ∑
2346:= 4
2286:= (
2144:− 1
1959:cos
1148:= 4
844:By
646:= (
445:If
97:or
65:of
63:set
57:In
3096::
3066:.
3036:.
3026:12
3024:.
2915:}}
2911:{{
2899:.
2887:11
2885:.
2881:.
2862:.
2807:MR
2805:.
2791:.
2785:.
2762:.
2754:.
2744:95
2742:.
2687:,
2655:,
2645:84
2643:,
2624:,
2620:,
2554:MR
2550:10
2548:,
2544:,
2334:−1
2321:−1
2303:−1
2295:−
2273:−1
2247:=
2236:−1
2212:+
2204:,
2191:−1
2157:/4
2082:.
2024:.
1158:A
1085:/4
1053:.
1038:.
965:,
961:,
957:,
947:BD
943:AC
856:,
852:,
658:+
654:+
650:+
635:,
631:,
627:,
534:.
515:.
460:.
400:.
229:,
217:.
187:.
166:PQ
154:OQ
150:OP
50:,
3086:.
3072:.
3044:.
3032::
3018:R
3006:.
2979:.
2952:.
2921:)
2903::
2893::
2857:"
2836:.
2832::
2813:.
2799::
2770:.
2750::
2726:.
2651::
2626:7
2603:.
2561:.
2527:.
2497:.
2479:/
2404:)
2401:k
2397:q
2392:k
2388:q
2383:k
2379:q
2377:2
2374:k
2369:A
2364:)
2361:k
2357:q
2352:k
2348:q
2343:k
2339:s
2332:n
2328:k
2323:)
2319:k
2315:c
2311:k
2307:c
2301:k
2297:c
2292:k
2288:c
2283:k
2279:q
2271:n
2267:c
2262:n
2258:q
2252:1
2249:c
2245:1
2242:q
2234:n
2230:θ
2226:1
2223:θ
2217:2
2214:θ
2210:1
2207:θ
2201:1
2198:θ
2189:n
2185:c
2181:1
2178:c
2172:n
2161:k
2154:k
2150:θ
2142:n
2135:n
2131:θ
2127:2
2124:θ
2115:1
2112:θ
2069:y
2065:x
1995:.
1986:=
1979:)
1974:n
1966:(
1955:1
1943:3
1940:=
1937:n
1881:.
1878:)
1873:4
1861:n
1857:A
1853:M
1846:+
1840:+
1835:4
1823:2
1819:A
1815:M
1808:+
1803:4
1791:1
1787:A
1783:M
1776:(
1773:n
1770:2
1767:=
1762:2
1758:)
1752:2
1740:n
1736:A
1732:M
1725:+
1719:+
1714:2
1702:2
1698:A
1694:M
1687:+
1682:2
1670:1
1666:A
1662:M
1655:(
1652:3
1638:i
1636:A
1632:M
1610:i
1606:A
1602:M
1589:n
1565:.
1557:n
1545:2
1539:/
1535:n
1521:1
1515:n
1511:A
1507:M
1501:+
1490:3
1484:n
1480:A
1476:M
1470:+
1464:+
1453:3
1449:A
1445:M
1439:+
1428:1
1424:A
1420:M
1410:;
1402:n
1390:2
1384:/
1380:n
1366:n
1362:A
1358:M
1352:+
1341:2
1335:n
1331:A
1327:M
1321:+
1315:+
1304:3
1300:A
1296:M
1290:+
1279:1
1275:A
1271:M
1262:{
1247:M
1241:n
1239:A
1236:1
1233:A
1228:M
1222:n
1220:A
1216:1
1213:A
1208:n
1196:n
1192:P
1188:n
1184:P
1180:P
1176:n
1172:n
1168:n
1153:n
1146:n
1141:n
1137:n
1083:π
1079:n
1032:X
1014:.
1011:D
1008:X
1002:X
999:B
996:=
993:C
990:X
984:X
981:A
967:D
963:C
959:B
955:A
951:X
926:.
923:D
920:A
914:C
911:B
908:+
905:D
902:C
896:B
893:A
890:=
887:D
884:B
878:C
875:A
862:D
858:C
854:B
850:A
822:,
815:)
812:d
806:s
803:(
800:)
797:c
791:s
788:(
785:)
782:b
776:s
773:(
770:)
767:a
761:s
758:(
753:)
750:c
747:b
744:+
741:d
738:a
735:(
732:)
729:d
726:b
723:+
720:c
717:a
714:(
711:)
708:d
705:c
702:+
699:b
696:a
693:(
684:4
681:1
676:=
673:R
660:d
656:c
652:b
648:a
644:s
637:d
633:c
629:b
625:a
603:D
600:B
597:C
591:=
588:D
585:A
582:C
499:T
475:T
381:.
374:)
371:c
365:b
362:+
359:a
356:(
353:)
350:c
347:+
344:b
338:a
335:(
332:)
329:c
326:+
323:b
320:+
317:a
311:(
308:)
305:c
302:+
299:b
296:+
293:a
290:(
283:2
279:c
273:2
269:b
263:2
259:a
251:=
248:R
235:c
231:b
227:a
185:O
179:n
177:(
175:n
170:n
158:O
146:Q
142:P
138:O
52:P
44:O
40:C
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
