3907:
2428:
479:
1972:
634:
601:
3894:
2491:
20:
793:
1962:
1785:
2413:
779:. In proposition 14 of book II Euclid gives a method for squaring a rectangle, which essentially matches the method given here. Euclid however provides a different slightly more complicated proof for the correctness of the construction rather than relying on the geometric mean theorem.
1386:
1797:
2154:
1587:
2165:
1138:
984:
1268:
853:
746:
76:
1278:
2170:
2011:
1957:{\displaystyle \tan \alpha \cdot \tan \beta ={\frac {h}{p}}\cdot {\frac {h}{q}}\,\implies \tan \alpha \cdot \cot \alpha ={\frac {h^{2}}{pq}}\implies 1={\frac {h^{2}}{pq}}\implies h={\sqrt {pq}}}
1592:
630:. Now the altitude represents the geometric mean and the radius the arithmetic mean of the two numbers. Since the altitude is always smaller or equal to the radius, this yields the inequality.
1524:
1185:
1034:
1780:{\displaystyle {\begin{aligned}\angle ACB&=\angle ACD+\angle DCB\\&=\angle ACD+(90^{\circ }-\angle DBC)\\&=\angle ACD+(90^{\circ }-\angle ACD)\\&=90^{\circ }\end{aligned}}}
2494:
Shear mappings with their associated fixed lines (dotted), starting with the original square as preimage each parallelogram displays the image of a shear mapping of the figure left of it
2006:
1568:
576:
2408:{\displaystyle {\begin{aligned}2h^{2}&=a^{2}+b^{2}-p^{2}-q^{2}\\&=c^{2}-p^{2}-q^{2}\\&=(p+q)^{2}-p^{2}-q^{2}\\&=2pq\\\therefore \ h^{2}&=pq.\end{aligned}}}
158:
389:
506:
1477:
1437:
429:
265:
200:
1052:
898:
206:
The converse statement is true as well. Any triangle, in which the altitude equals the geometric mean of the two line segments created by it, is a right triangle.
452:
2437:
yields two similar triangles, which can be augmented and arranged in two alternative ways into a larger right triangle with perpendicular sides of lengths
2663:
1211:
3767:
1581:
have an angle of equal size and have corresponding pairs of legs with the same ratio. This means the triangles are similar, which yields:
3367:
799:
1143:
992:
3845:
3148:
2684:
3946:
2656:
3692:
645:
1381:{\displaystyle {\frac {h}{p}}={\frac {q}{h}}\,\Leftrightarrow \,h^{2}=pq\,\Leftrightarrow \,h={\sqrt {pq}}\qquad (h,p,q>0)}
3422:
2612:
2558:
2531:
267:
version of the formula yields a method to construct a square of equal area to a given rectangle through the following steps:
220:
26:
3387:
3755:
3158:
3822:
2649:
1482:
1272:
Because of the similarity we get the following equality of ratios and its algebraic rearrangement yields the theorem:
3936:
2149:{\displaystyle {\begin{aligned}h^{2}&=a^{2}-q^{2}\\h^{2}&=b^{2}-p^{2}\\c^{2}&=a^{2}+b^{2}\end{aligned}}}
2467:. Since both arrangements yield the same triangle, the areas of the square and the rectangle must be identical.
3791:
3724:
3357:
3237:
3941:
3860:
3617:
3505:
2680:
1529:
545:
514:
to 1 (note some letters are different than used throughout the rest of the article, notably the articles'
3569:
3500:
752:
638:
3196:
3012:
2987:
3731:
3702:
3062:
2917:
877:
130:
3865:
3599:
3142:
365:
3827:
3803:
3677:
3612:
3553:
3490:
3480:
3216:
3135:
2997:
2907:
2787:
487:
3097:
3850:
3798:
3697:
3525:
3460:
3455:
3226:
3027:
2962:
2952:
2902:
2574:
2630:
1446:
1406:
398:
234:
169:
3898:
3750:
3594:
3535:
3412:
3335:
3283:
3085:
2992:
2842:
2602:
2548:
2521:
3875:
3815:
3779:
3622:
3445:
3392:
3362:
3352:
3261:
3124:
3017:
2932:
2887:
2867:
2712:
2697:
776:
609:
535:
8:
3855:
3774:
3762:
3743:
3707:
3627:
3545:
3530:
3520:
3470:
3465:
3407:
3276:
3168:
3032:
3022:
2922:
2892:
2832:
2807:
2732:
2722:
2707:
2475:
The square of the altitude can be transformed into an rectangle of equal area with sides
94:
434:
3911:
3870:
3810:
3738:
3604:
3579:
3397:
3372:
3340:
3178:
2937:
2882:
2847:
2742:
756:
343:
82:
124:
denote the segments that the altitude creates on the hypotenuse, it can be stated as:
3906:
3584:
3495:
3345:
3271:
3244:
3007:
2827:
2817:
2752:
2672:
2608:
2554:
2527:
2427:
291:
478:
3786:
3657:
3574:
3330:
3318:
3266:
3002:
775:(ca. 360–280 BC), who stated it as a corollary to proposition 8 in book VI of his
3427:
3417:
3311:
3037:
759:
ensures that the hypotenuse of the right angled triangle is the diameter of its
3687:
3682:
3510:
3402:
3382:
3210:
2767:
2737:
1971:
392:
106:
102:
1133:{\displaystyle \angle ACB=\angle BDC=90^{\circ },\quad \angle ABC=\angle CBD;}
979:{\displaystyle \angle ACB=\angle ADC=90^{\circ },\quad \angle BAC=\angle CAD;}
3931:
3925:
3647:
3515:
3485:
3306:
3114:
3057:
2598:
2484:
1979:
In the setting of the geometric mean theorem there are three right triangles
2641:
3589:
3377:
3052:
2812:
2802:
2636:
987:
760:
105:
and the two line segments it creates on the hypotenuse. It states that the
3203:
3091:
2797:
2782:
2159:
Adding the first 2 two equations and then using the third then leads to:
359:
633:
3189:
3108:
3047:
3042:
2982:
2967:
2912:
2897:
2852:
2792:
2777:
2757:
2727:
2692:
608:
Another application of the theorem provides a geometrical proof of the
334:
as its diameter. Then we erect a perpendicular line to the diameter in
98:
431:
directly shows that a square with the area of the rectangle (equal to
2942:
2927:
2877:
2772:
2762:
2747:
2717:
600:
3079:
2857:
2702:
2490:
1263:{\displaystyle \triangle ACD\sim \triangle ABC\sim \triangle BCD.}
3652:
2977:
2972:
2872:
2862:
2837:
848:{\displaystyle \triangle ABC\sim \triangle ADC\sim \triangle DBC}
2418:
which finally yields the formula of the geometric mean theorem.
2947:
2822:
772:
19:
3323:
3301:
2957:
792:
751:
The theorem can also be thought of as a special case of the
542:
is 1, since then the first version of the formula becomes
2550:
Icons of
Mathematics: An Exploration of Twenty Key Images
741:{\displaystyle |CD||DE|=|AD||DB|\Leftrightarrow h^{2}=pq}
2421:
214:
1502:
1487:
534:
also allows for the construction of square roots (see
71:{\displaystyle h^{2}=pq\Leftrightarrow h={\sqrt {pq}}}
2168:
2009:
1800:
1590:
1532:
1485:
1449:
1409:
1281:
1214:
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1055:
995:
901:
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648:
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490:
437:
401:
368:
237:
172:
133:
29:
1966:
2407:
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1956:
1779:
1562:
1518:
1471:
1431:
1380:
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1179:
1132:
1028:
978:
847:
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570:
500:
446:
423:
383:
259:
194:
152:
70:
2597:
2520:Wellstein, Hartmut; Kirsche, Peter (2009-06-16).
2457:. One such arrangement requires a square of area
2433:Dissecting the right triangle along its altitude
3923:
2519:
1519:{\displaystyle {\tfrac {h}{p}}={\tfrac {q}{h}}.}
1180:{\displaystyle \triangle ABC\sim \triangle BCD.}
1029:{\displaystyle \triangle ABC\sim \triangle ACD.}
637:geometric mean theorem as a special case of the
2547:Alsina, Claudi; Nelsen, Roger B. (2011-08-04).
1790:
595:
2463:to complete it, the other a rectangle of area
23:area of grey square = area of grey rectangle:
2671:
2657:
473:
116:denotes the altitude in a right triangle and
2546:
612:in the case of two numbers. For the numbers
588:in the formula) will readily be the root of
620:one constructs a half circle with diameter
395:, and so the theorem applies: its identity
2664:
2650:
2470:
1937:
1933:
1904:
1900:
1853:
1849:
538:) by starting with a rectangle whose side
279:depicts the vertices and arches mentioned)
2000:in which the Pythagorean theorem yields:
1848:
1439:holds and need to show that the angle at
1333:
1329:
1309:
1305:
531:
109:of the two segments equals the altitude.
2489:
1970:
791:
632:
599:
477:
316:) and draw a half circle with endpoints
272:
112:Expressed as a mathematical formula, if
18:
226:
3924:
3693:Latin translations of the 12th century
787:
532:above method from Squaring a rectangle
221:straightedge and compass constructions
3423:Straightedge and compass construction
2645:
2422:Based on dissection and rearrangement
1563:{\displaystyle \angle ADC=\angle CDB}
771:The theorem is usually attributed to
571:{\displaystyle h={\sqrt {p\times 1}}}
219:The theorem is used in the following
215:Straightedge and compass construction
3388:Incircle and excircles of a triangle
2515:
2513:
2511:
2509:
2507:
2487:(shear mappings preserve the area):
1396:For the converse we have a triangle
755:for a circle, since the converse of
338:that intersects the half circle in
13:
1975:Proof with the Pythagorean theorem
1738:
1707:
1682:
1651:
1629:
1614:
1595:
1548:
1533:
1245:
1230:
1215:
1162:
1147:
1115:
1100:
1071:
1056:
1011:
996:
961:
946:
917:
902:
833:
818:
803:
369:
14:
3958:
2624:
2575:"Euclid's Elements, Introduction"
2504:
1443:is a right angle. Now because of
3905:
3892:
2426:
1967:Based on the Pythagorean theorem
454:) can be drawn by using exactly
1350:
1099:
945:
209:
91:right triangle altitude theorem
3947:Theorems about right triangles
3725:A History of Greek Mathematics
3238:The Quadrature of the Parabola
2591:
2567:
2540:
2526:(in German). Springer-Verlag.
2311:
2298:
1934:
1901:
1850:
1750:
1722:
1694:
1666:
1375:
1351:
1330:
1306:
1140:therefore by the AA postulate
716:
712:
701:
696:
685:
677:
666:
661:
650:
460:as the squares' side, because
275:Proof > Based on similarity
153:{\displaystyle h={\sqrt {pq}}}
49:
1:
2607:. American Mathematical Soc.
2498:
384:{\displaystyle \triangle ABC}
16:Theorem about right triangles
3506:Intersecting secants theorem
2601:; Friedrich, Thomas (2008).
1791:Based on trigonometric ratio
1038:further, consider triangles
596:Relation with other theorems
298:. Now we extend the segment
7:
3501:Intersecting chords theorem
3368:Doctrine of proportionality
753:intersecting chords theorem
501:{\displaystyle {\sqrt {p}}}
282:For a rectangle with sides
10:
3963:
3197:On the Sphere and Cylinder
3150:On the Sizes and Distances
1190:Therefore, both triangles
766:
474:Constructing a square root
93:is a relation between the
3899:Ancient Greece portal
3888:
3838:
3716:
3703:Philosophy of mathematics
3673:
3666:
3640:
3618:Ptolemy's table of chords
3562:
3544:
3443:
3436:
3292:
3254:
3071:
2679:
2673:Ancient Greek mathematics
3937:Euclidean plane geometry
3570:Aristarchus's inequality
3143:On Conoids and Spheroids
1472:{\displaystyle h^{2}=pq}
1432:{\displaystyle h^{2}=pq}
782:
424:{\displaystyle h^{2}=pq}
260:{\displaystyle h^{2}=pq}
195:{\displaystyle h^{2}=pq}
3678:Ancient Greek astronomy
3491:Inscribed angle theorem
3481:Greek geometric algebra
3136:Measurement of a Circle
2483:with the help of three
2471:Based on shear mappings
290:we denote its top left
3912:Mathematics portal
3698:Non-Euclidean geometry
3653:Mouseion of Alexandria
3526:Tangent-secant theorem
3476:Geometric mean theorem
3461:Exterior angle theorem
3456:Angle bisector theorem
3160:On Sizes and Distances
2495:
2409:
2150:
1976:
1958:
1781:
1564:
1520:
1473:
1433:
1382:
1264:
1181:
1134:
1030:
980:
856:
849:
748:
742:
605:
572:
527:
502:
448:
425:
385:
261:
196:
154:
87:geometric mean theorem
78:
72:
3600:Pappus's area theorem
3536:Theorem of the gnomon
3413:Quadratrix of Hippias
3336:Circles of Apollonius
3284:Problem of Apollonius
3262:Constructible numbers
3086:Archimedes Palimpsest
2493:
2410:
2151:
1974:
1959:
1782:
1565:
1521:
1474:
1434:
1383:
1265:
1208:and themselves, i.e.
1182:
1135:
1031:
981:
850:
795:
743:
636:
603:
573:
503:
481:
449:
426:
386:
324:with the new segment
262:
197:
163:or in term of areas:
155:
73:
22:
3816:prehistoric counting
3613:Ptolemy's inequality
3554:Apollonius's theorem
3393:Method of exhaustion
3363:Diophantine equation
3353:Circumscribed circle
3170:On the Moving Sphere
2166:
2007:
1798:
1588:
1530:
1483:
1447:
1407:
1279:
1212:
1144:
1053:
993:
899:
800:
646:
546:
536:constructible number
488:
435:
399:
366:
235:
227:Squaring a rectangle
170:
131:
27:
3942:History of geometry
3902: •
3708:Neusis construction
3628:Spiral of Theodorus
3521:Pythagorean theorem
3466:Euclidean algorithm
3408:Lune of Hippocrates
3277:Squaring the circle
3033:Theon of Alexandria
2708:Aristaeus the Elder
2604:Elementary Geometry
884:consider triangles
788:Based on similarity
3595:Menelaus's theorem
3585:Irrational numbers
3398:Parallel postulate
3373:Euclidean geometry
3341:Apollonian circles
2883:Isidore of Miletus
2523:Elementargeometrie
2496:
2405:
2403:
2146:
2144:
1977:
1954:
1777:
1775:
1560:
1516:
1511:
1496:
1469:
1429:
1392:Proof of converse:
1378:
1260:
1177:
1130:
1026:
976:
857:
845:
749:
738:
606:
568:
528:
498:
447:{\displaystyle pq}
444:
421:
381:
346:the angle between
257:
192:
150:
83:Euclidean geometry
79:
68:
3919:
3918:
3884:
3883:
3636:
3635:
3623:Ptolemy's theorem
3496:Intercept theorem
3346:Apollonian gasket
3272:Doubling the cube
3245:The Sand Reckoner
2614:978-0-8218-4347-5
2579:aleph0.clarku.edu
2560:978-0-88385-352-8
2533:978-3-8348-0856-1
2374:
1952:
1931:
1898:
1846:
1833:
1510:
1495:
1348:
1303:
1290:
986:therefore by the
566:
496:
148:
66:
3954:
3910:
3909:
3897:
3896:
3895:
3671:
3670:
3658:Platonic Academy
3605:Problem II.8 of
3575:Crossbar theorem
3531:Thales's theorem
3471:Euclid's theorem
3441:
3440:
3358:Commensurability
3319:Axiomatic system
3267:Angle trisection
3232:
3222:
3184:
3174:
3164:
3154:
3130:
3120:
3103:
2666:
2659:
2652:
2643:
2642:
2619:
2618:
2595:
2589:
2588:
2586:
2585:
2571:
2565:
2564:
2544:
2538:
2537:
2517:
2482:
2478:
2466:
2462:
2456:
2446:
2436:
2430:
2414:
2412:
2411:
2406:
2404:
2384:
2383:
2372:
2349:
2345:
2344:
2332:
2331:
2319:
2318:
2291:
2287:
2286:
2274:
2273:
2261:
2260:
2245:
2241:
2240:
2228:
2227:
2215:
2214:
2202:
2201:
2185:
2184:
2155:
2153:
2152:
2147:
2145:
2141:
2140:
2128:
2127:
2111:
2110:
2097:
2096:
2084:
2083:
2067:
2066:
2053:
2052:
2040:
2039:
2023:
2022:
1999:
1992:
1985:
1963:
1961:
1960:
1955:
1953:
1945:
1932:
1930:
1922:
1921:
1912:
1899:
1897:
1889:
1888:
1879:
1847:
1839:
1834:
1826:
1786:
1784:
1783:
1778:
1776:
1772:
1771:
1756:
1734:
1733:
1700:
1678:
1677:
1644:
1580:
1569:
1567:
1566:
1561:
1525:
1523:
1522:
1517:
1512:
1503:
1497:
1488:
1478:
1476:
1475:
1470:
1459:
1458:
1442:
1438:
1436:
1435:
1430:
1419:
1418:
1402:
1387:
1385:
1384:
1379:
1349:
1341:
1319:
1318:
1304:
1296:
1291:
1283:
1269:
1267:
1266:
1261:
1207:
1200:
1186:
1184:
1183:
1178:
1139:
1137:
1136:
1131:
1095:
1094:
1048:
1035:
1033:
1032:
1027:
985:
983:
982:
977:
941:
940:
894:
875:
860:Proof of theorem
854:
852:
851:
846:
747:
745:
744:
739:
728:
727:
715:
704:
699:
688:
680:
669:
664:
653:
629:
619:
615:
610:AM–GM inequality
604:AM-GM inequality
591:
587:
583:
582:
577:
575:
574:
569:
567:
556:
541:
525:
524:
519:
518:
513:
509:
507:
505:
504:
499:
497:
492:
482:Construction of
469:
465:
464:
459:
458:
453:
451:
450:
445:
430:
428:
427:
422:
411:
410:
390:
388:
387:
382:
357:
356:
351:
350:
341:
337:
333:
323:
319:
315:
311:
310:
305:
301:
297:
289:
285:
266:
264:
263:
258:
247:
246:
201:
199:
198:
193:
182:
181:
159:
157:
156:
151:
149:
141:
123:
119:
115:
77:
75:
74:
69:
67:
59:
39:
38:
3962:
3961:
3957:
3956:
3955:
3953:
3952:
3951:
3922:
3921:
3920:
3915:
3904:
3893:
3891:
3880:
3846:Arabian/Islamic
3834:
3823:numeral systems
3712:
3662:
3632:
3580:Heron's formula
3558:
3540:
3432:
3428:Triangle center
3418:Regular polygon
3295:and definitions
3294:
3288:
3250:
3230:
3220:
3182:
3172:
3162:
3152:
3128:
3118:
3101:
3067:
3038:Theon of Smyrna
2683:
2675:
2670:
2627:
2622:
2615:
2596:
2592:
2583:
2581:
2573:
2572:
2568:
2561:
2545:
2541:
2534:
2518:
2505:
2501:
2480:
2476:
2473:
2464:
2458:
2448:
2438:
2434:
2424:
2402:
2401:
2385:
2379:
2375:
2366:
2365:
2347:
2346:
2340:
2336:
2327:
2323:
2314:
2310:
2289:
2288:
2282:
2278:
2269:
2265:
2256:
2252:
2243:
2242:
2236:
2232:
2223:
2219:
2210:
2206:
2197:
2193:
2186:
2180:
2176:
2169:
2167:
2164:
2163:
2143:
2142:
2136:
2132:
2123:
2119:
2112:
2106:
2102:
2099:
2098:
2092:
2088:
2079:
2075:
2068:
2062:
2058:
2055:
2054:
2048:
2044:
2035:
2031:
2024:
2018:
2014:
2010:
2008:
2005:
2004:
1994:
1987:
1980:
1969:
1944:
1923:
1917:
1913:
1911:
1890:
1884:
1880:
1878:
1838:
1825:
1799:
1796:
1795:
1793:
1774:
1773:
1767:
1763:
1754:
1753:
1729:
1725:
1698:
1697:
1673:
1669:
1642:
1641:
1607:
1591:
1589:
1586:
1585:
1571:
1531:
1528:
1527:
1501:
1486:
1484:
1481:
1480:
1454:
1450:
1448:
1445:
1444:
1440:
1414:
1410:
1408:
1405:
1404:
1397:
1340:
1314:
1310:
1295:
1282:
1280:
1277:
1276:
1213:
1210:
1209:
1202:
1201:are similar to
1191:
1145:
1142:
1141:
1090:
1086:
1054:
1051:
1050:
1049:; here we have
1039:
994:
991:
990:
936:
932:
900:
897:
896:
895:; here we have
885:
866:
855:
801:
798:
797:
790:
785:
769:
757:Thales' theorem
723:
719:
711:
700:
695:
684:
676:
665:
660:
649:
647:
644:
643:
642:
621:
617:
613:
598:
589:
585:
580:
579:
578:, showing that
555:
547:
544:
543:
539:
522:
521:
516:
515:
511:
491:
489:
486:
485:
483:
476:
467:
462:
461:
456:
455:
436:
433:
432:
406:
402:
400:
397:
396:
367:
364:
363:
354:
353:
348:
347:
344:Thales' theorem
339:
335:
325:
321:
317:
313:
308:
307:
303:
302:to its left by
299:
295:
287:
283:
281:
242:
238:
236:
233:
232:
229:
217:
212:
177:
173:
171:
168:
167:
140:
132:
129:
128:
121:
117:
113:
58:
34:
30:
28:
25:
24:
17:
12:
11:
5:
3960:
3950:
3949:
3944:
3939:
3934:
3917:
3916:
3889:
3886:
3885:
3882:
3881:
3879:
3878:
3873:
3868:
3863:
3858:
3853:
3848:
3842:
3840:
3839:Other cultures
3836:
3835:
3833:
3832:
3831:
3830:
3820:
3819:
3818:
3808:
3807:
3806:
3796:
3795:
3794:
3784:
3783:
3782:
3772:
3771:
3770:
3760:
3759:
3758:
3748:
3747:
3746:
3736:
3735:
3734:
3720:
3718:
3714:
3713:
3711:
3710:
3705:
3700:
3695:
3690:
3688:Greek numerals
3685:
3683:Attic numerals
3680:
3674:
3668:
3664:
3663:
3661:
3660:
3655:
3650:
3644:
3642:
3638:
3637:
3634:
3633:
3631:
3630:
3625:
3620:
3615:
3610:
3602:
3597:
3592:
3587:
3582:
3577:
3572:
3566:
3564:
3560:
3559:
3557:
3556:
3550:
3548:
3542:
3541:
3539:
3538:
3533:
3528:
3523:
3518:
3513:
3511:Law of cosines
3508:
3503:
3498:
3493:
3488:
3483:
3478:
3473:
3468:
3463:
3458:
3452:
3450:
3438:
3434:
3433:
3431:
3430:
3425:
3420:
3415:
3410:
3405:
3403:Platonic solid
3400:
3395:
3390:
3385:
3383:Greek numerals
3380:
3375:
3370:
3365:
3360:
3355:
3350:
3349:
3348:
3343:
3333:
3328:
3327:
3326:
3316:
3315:
3314:
3309:
3298:
3296:
3290:
3289:
3287:
3286:
3281:
3280:
3279:
3274:
3269:
3258:
3256:
3252:
3251:
3249:
3248:
3241:
3234:
3224:
3214:
3211:Planisphaerium
3207:
3200:
3193:
3186:
3176:
3166:
3156:
3146:
3139:
3132:
3122:
3112:
3105:
3095:
3088:
3083:
3075:
3073:
3069:
3068:
3066:
3065:
3060:
3055:
3050:
3045:
3040:
3035:
3030:
3025:
3020:
3015:
3010:
3005:
3000:
2995:
2990:
2985:
2980:
2975:
2970:
2965:
2960:
2955:
2950:
2945:
2940:
2935:
2930:
2925:
2920:
2915:
2910:
2905:
2900:
2895:
2890:
2885:
2880:
2875:
2870:
2865:
2860:
2855:
2850:
2845:
2840:
2835:
2830:
2825:
2820:
2815:
2810:
2805:
2800:
2795:
2790:
2785:
2780:
2775:
2770:
2765:
2760:
2755:
2750:
2745:
2740:
2735:
2730:
2725:
2720:
2715:
2710:
2705:
2700:
2695:
2689:
2687:
2681:Mathematicians
2677:
2676:
2669:
2668:
2661:
2654:
2646:
2640:
2639:
2632:Geometric Mean
2626:
2625:External links
2623:
2621:
2620:
2613:
2599:Agricola, Ilka
2590:
2566:
2559:
2539:
2532:
2502:
2500:
2497:
2485:shear mappings
2472:
2469:
2423:
2420:
2416:
2415:
2400:
2397:
2394:
2391:
2388:
2386:
2382:
2378:
2371:
2368:
2367:
2364:
2361:
2358:
2355:
2352:
2350:
2348:
2343:
2339:
2335:
2330:
2326:
2322:
2317:
2313:
2309:
2306:
2303:
2300:
2297:
2294:
2292:
2290:
2285:
2281:
2277:
2272:
2268:
2264:
2259:
2255:
2251:
2248:
2246:
2244:
2239:
2235:
2231:
2226:
2222:
2218:
2213:
2209:
2205:
2200:
2196:
2192:
2189:
2187:
2183:
2179:
2175:
2172:
2171:
2157:
2156:
2139:
2135:
2131:
2126:
2122:
2118:
2115:
2113:
2109:
2105:
2101:
2100:
2095:
2091:
2087:
2082:
2078:
2074:
2071:
2069:
2065:
2061:
2057:
2056:
2051:
2047:
2043:
2038:
2034:
2030:
2027:
2025:
2021:
2017:
2013:
2012:
1968:
1965:
1951:
1948:
1943:
1940:
1936:
1929:
1926:
1920:
1916:
1910:
1907:
1903:
1896:
1893:
1887:
1883:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1852:
1845:
1842:
1837:
1832:
1829:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1792:
1789:
1788:
1787:
1770:
1766:
1762:
1759:
1757:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1732:
1728:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1701:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1676:
1672:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1645:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1608:
1606:
1603:
1600:
1597:
1594:
1593:
1570:the triangles
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1526:Together with
1515:
1509:
1506:
1500:
1494:
1491:
1468:
1465:
1462:
1457:
1453:
1428:
1425:
1422:
1417:
1413:
1389:
1388:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1347:
1344:
1339:
1336:
1332:
1328:
1325:
1322:
1317:
1313:
1308:
1302:
1299:
1294:
1289:
1286:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1188:
1187:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1098:
1093:
1089:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1036:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
944:
939:
935:
931:
928:
925:
922:
919:
916:
913:
910:
907:
904:
865:The triangles
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
808:
805:
796:
789:
786:
784:
781:
768:
765:
737:
734:
731:
726:
722:
718:
714:
710:
707:
703:
698:
694:
691:
687:
683:
679:
675:
672:
668:
663:
659:
656:
652:
597:
594:
565:
562:
559:
554:
551:
495:
475:
472:
443:
440:
420:
417:
414:
409:
405:
393:right triangle
380:
377:
374:
371:
271:(The image in
256:
253:
250:
245:
241:
228:
225:
216:
213:
211:
208:
204:
203:
191:
188:
185:
180:
176:
161:
160:
147:
144:
139:
136:
107:geometric mean
103:right triangle
65:
62:
57:
54:
51:
48:
45:
42:
37:
33:
15:
9:
6:
4:
3:
2:
3959:
3948:
3945:
3943:
3940:
3938:
3935:
3933:
3930:
3929:
3927:
3914:
3913:
3908:
3901:
3900:
3887:
3877:
3874:
3872:
3869:
3867:
3864:
3862:
3859:
3857:
3854:
3852:
3849:
3847:
3844:
3843:
3841:
3837:
3829:
3826:
3825:
3824:
3821:
3817:
3814:
3813:
3812:
3809:
3805:
3802:
3801:
3800:
3797:
3793:
3790:
3789:
3788:
3785:
3781:
3778:
3777:
3776:
3773:
3769:
3766:
3765:
3764:
3761:
3757:
3754:
3753:
3752:
3749:
3745:
3742:
3741:
3740:
3737:
3733:
3729:
3728:
3727:
3726:
3722:
3721:
3719:
3715:
3709:
3706:
3704:
3701:
3699:
3696:
3694:
3691:
3689:
3686:
3684:
3681:
3679:
3676:
3675:
3672:
3669:
3665:
3659:
3656:
3654:
3651:
3649:
3646:
3645:
3643:
3639:
3629:
3626:
3624:
3621:
3619:
3616:
3614:
3611:
3609:
3608:
3603:
3601:
3598:
3596:
3593:
3591:
3588:
3586:
3583:
3581:
3578:
3576:
3573:
3571:
3568:
3567:
3565:
3561:
3555:
3552:
3551:
3549:
3547:
3543:
3537:
3534:
3532:
3529:
3527:
3524:
3522:
3519:
3517:
3516:Pons asinorum
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3492:
3489:
3487:
3486:Hinge theorem
3484:
3482:
3479:
3477:
3474:
3472:
3469:
3467:
3464:
3462:
3459:
3457:
3454:
3453:
3451:
3449:
3448:
3442:
3439:
3435:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3384:
3381:
3379:
3376:
3374:
3371:
3369:
3366:
3364:
3361:
3359:
3356:
3354:
3351:
3347:
3344:
3342:
3339:
3338:
3337:
3334:
3332:
3329:
3325:
3322:
3321:
3320:
3317:
3313:
3310:
3308:
3305:
3304:
3303:
3300:
3299:
3297:
3291:
3285:
3282:
3278:
3275:
3273:
3270:
3268:
3265:
3264:
3263:
3260:
3259:
3257:
3253:
3247:
3246:
3242:
3240:
3239:
3235:
3233:
3229:
3225:
3223:
3219:
3215:
3213:
3212:
3208:
3206:
3205:
3201:
3199:
3198:
3194:
3192:
3191:
3187:
3185:
3181:
3177:
3175:
3171:
3167:
3165:
3161:
3157:
3155:
3153:(Aristarchus)
3151:
3147:
3145:
3144:
3140:
3138:
3137:
3133:
3131:
3127:
3123:
3121:
3117:
3113:
3111:
3110:
3106:
3104:
3100:
3096:
3094:
3093:
3089:
3087:
3084:
3082:
3081:
3077:
3076:
3074:
3070:
3064:
3061:
3059:
3058:Zeno of Sidon
3056:
3054:
3051:
3049:
3046:
3044:
3041:
3039:
3036:
3034:
3031:
3029:
3026:
3024:
3021:
3019:
3016:
3014:
3011:
3009:
3006:
3004:
3001:
2999:
2996:
2994:
2991:
2989:
2986:
2984:
2981:
2979:
2976:
2974:
2971:
2969:
2966:
2964:
2961:
2959:
2956:
2954:
2951:
2949:
2946:
2944:
2941:
2939:
2936:
2934:
2931:
2929:
2926:
2924:
2921:
2919:
2916:
2914:
2911:
2909:
2906:
2904:
2901:
2899:
2896:
2894:
2891:
2889:
2886:
2884:
2881:
2879:
2876:
2874:
2871:
2869:
2866:
2864:
2861:
2859:
2856:
2854:
2851:
2849:
2846:
2844:
2841:
2839:
2836:
2834:
2831:
2829:
2826:
2824:
2821:
2819:
2816:
2814:
2811:
2809:
2806:
2804:
2801:
2799:
2796:
2794:
2791:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2764:
2761:
2759:
2756:
2754:
2751:
2749:
2746:
2744:
2741:
2739:
2736:
2734:
2731:
2729:
2726:
2724:
2721:
2719:
2716:
2714:
2711:
2709:
2706:
2704:
2701:
2699:
2696:
2694:
2691:
2690:
2688:
2686:
2682:
2678:
2674:
2667:
2662:
2660:
2655:
2653:
2648:
2647:
2644:
2638:
2634:
2633:
2629:
2628:
2616:
2610:
2606:
2605:
2600:
2594:
2580:
2576:
2570:
2562:
2556:
2552:
2551:
2543:
2535:
2529:
2525:
2524:
2516:
2514:
2512:
2510:
2508:
2503:
2492:
2488:
2486:
2468:
2461:
2455:
2451:
2445:
2441:
2431:
2429:
2419:
2398:
2395:
2392:
2389:
2387:
2380:
2376:
2369:
2362:
2359:
2356:
2353:
2351:
2341:
2337:
2333:
2328:
2324:
2320:
2315:
2307:
2304:
2301:
2295:
2293:
2283:
2279:
2275:
2270:
2266:
2262:
2257:
2253:
2249:
2247:
2237:
2233:
2229:
2224:
2220:
2216:
2211:
2207:
2203:
2198:
2194:
2190:
2188:
2181:
2177:
2173:
2162:
2161:
2160:
2137:
2133:
2129:
2124:
2120:
2116:
2114:
2107:
2103:
2093:
2089:
2085:
2080:
2076:
2072:
2070:
2063:
2059:
2049:
2045:
2041:
2036:
2032:
2028:
2026:
2019:
2015:
2003:
2002:
2001:
1998:
1991:
1984:
1973:
1964:
1949:
1946:
1941:
1938:
1927:
1924:
1918:
1914:
1908:
1905:
1894:
1891:
1885:
1881:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1843:
1840:
1835:
1830:
1827:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1768:
1764:
1760:
1758:
1747:
1744:
1741:
1735:
1730:
1726:
1719:
1716:
1713:
1710:
1704:
1702:
1691:
1688:
1685:
1679:
1674:
1670:
1663:
1660:
1657:
1654:
1648:
1646:
1638:
1635:
1632:
1626:
1623:
1620:
1617:
1611:
1609:
1604:
1601:
1598:
1584:
1583:
1582:
1579:
1575:
1557:
1554:
1551:
1545:
1542:
1539:
1536:
1513:
1507:
1504:
1498:
1492:
1489:
1479:we also have
1466:
1463:
1460:
1455:
1451:
1426:
1423:
1420:
1415:
1411:
1401:
1394:
1393:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1345:
1342:
1337:
1334:
1326:
1323:
1320:
1315:
1311:
1300:
1297:
1292:
1287:
1284:
1275:
1274:
1273:
1270:
1257:
1254:
1251:
1248:
1242:
1239:
1236:
1233:
1227:
1224:
1221:
1218:
1206:
1199:
1195:
1174:
1171:
1168:
1165:
1159:
1156:
1153:
1150:
1127:
1124:
1121:
1118:
1112:
1109:
1106:
1103:
1096:
1091:
1087:
1083:
1080:
1077:
1074:
1068:
1065:
1062:
1059:
1047:
1043:
1037:
1023:
1020:
1017:
1014:
1008:
1005:
1002:
999:
989:
973:
970:
967:
964:
958:
955:
952:
949:
942:
937:
933:
929:
926:
923:
920:
914:
911:
908:
905:
893:
889:
883:
882:
881:
879:
874:
870:
863:
861:
842:
839:
836:
830:
827:
824:
821:
815:
812:
809:
806:
794:
780:
778:
774:
764:
762:
758:
754:
735:
732:
729:
724:
720:
708:
705:
692:
689:
681:
673:
670:
657:
654:
640:
639:chord theorem
635:
631:
628:
624:
611:
602:
593:
563:
560:
557:
552:
549:
537:
533:
526:on the image)
493:
480:
471:
441:
438:
418:
415:
412:
407:
403:
394:
378:
375:
372:
361:
345:
332:
328:
293:
280:
278:
276:
268:
254:
251:
248:
243:
239:
224:
222:
207:
189:
186:
183:
178:
174:
166:
165:
164:
145:
142:
137:
134:
127:
126:
125:
110:
108:
104:
100:
96:
92:
88:
84:
63:
60:
55:
52:
46:
43:
40:
35:
31:
21:
3903:
3890:
3732:Thomas Heath
3723:
3606:
3590:Law of sines
3475:
3446:
3378:Golden ratio
3243:
3236:
3227:
3221:(Theodosius)
3217:
3209:
3202:
3195:
3188:
3179:
3169:
3163:(Hipparchus)
3159:
3149:
3141:
3134:
3125:
3115:
3107:
3102:(Apollonius)
3098:
3090:
3078:
3053:Zeno of Elea
2813:Eratosthenes
2803:Dionysodorus
2637:Cut-the-Knot
2631:
2603:
2593:
2582:. Retrieved
2578:
2569:
2549:
2542:
2522:
2474:
2459:
2453:
2449:
2443:
2439:
2432:
2425:
2417:
2158:
1996:
1989:
1982:
1978:
1794:
1577:
1573:
1399:
1395:
1391:
1390:
1271:
1204:
1197:
1193:
1189:
1045:
1041:
988:AA postulate
891:
887:
872:
868:
864:
859:
858:
770:
761:circumcircle
750:
626:
622:
607:
529:
330:
326:
312:centered on
274:
270:
269:
230:
218:
210:Applications
205:
162:
111:
90:
86:
80:
3799:mathematics
3607:Arithmetica
3204:Ostomachion
3173:(Autolycus)
3092:Arithmetica
2868:Hippocrates
2798:Dinostratus
2783:Dicaearchus
2713:Aristarchus
510:by setting
360:right angle
306:(using arc
3926:Categories
3851:Babylonian
3751:arithmetic
3717:History of
3546:Apollonius
3231:(Menelaus)
3190:On Spirals
3109:Catoptrics
3048:Xenocrates
3043:Thymaridas
3028:Theodosius
3013:Theaetetus
2993:Simplicius
2983:Pythagoras
2968:Posidonius
2953:Philonides
2913:Nicomachus
2908:Metrodorus
2898:Menaechmus
2853:Hipparchus
2843:Heliodorus
2793:Diophantus
2778:Democritus
2758:Chrysippus
2728:Archimedes
2723:Apollonius
2693:Anaxagoras
2685:(timeline)
2584:2024-09-17
2499:References
99:hypotenuse
3312:Inscribed
3072:Treatises
3063:Zenodorus
3023:Theodorus
2998:Sosigenes
2943:Philolaus
2928:Oenopides
2923:Nicoteles
2918:Nicomedes
2878:Hypsicles
2773:Ctesibius
2763:Cleomedes
2748:Callippus
2733:Autolycus
2718:Aristotle
2698:Anthemius
2370:∴
2334:−
2321:−
2276:−
2263:−
2230:−
2217:−
2086:−
2042:−
1935:⟹
1902:⟹
1873:α
1870:
1864:⋅
1861:α
1858:
1851:⟹
1836:⋅
1820:β
1817:
1811:⋅
1808:α
1805:
1769:∘
1739:∠
1736:−
1731:∘
1708:∠
1683:∠
1680:−
1675:∘
1652:∠
1630:∠
1615:∠
1596:∠
1549:∠
1534:∠
1403:in which
1331:⇔
1307:⇔
1246:△
1243:∼
1231:△
1228:∼
1216:△
1163:△
1160:∼
1148:△
1116:∠
1101:∠
1092:∘
1072:∠
1057:∠
1012:△
1009:∼
997:△
962:∠
947:∠
938:∘
918:∠
903:∠
880:, since:
834:△
831:∼
819:△
816:∼
804:△
717:⇔
561:×
370:△
342:. As per
50:⇔
3876:Japanese
3861:Egyptian
3804:timeline
3792:timeline
3780:timeline
3775:geometry
3768:timeline
3763:calculus
3756:timeline
3744:timeline
3447:Elements
3293:Concepts
3255:Problems
3228:Spherics
3218:Spherics
3183:(Euclid)
3129:(Euclid)
3126:Elements
3119:(Euclid)
3080:Almagest
2988:Serenus
2963:Porphyry
2903:Menelaus
2858:Hippasus
2833:Eutocius
2808:Domninus
2703:Archytas
777:Elements
95:altitude
3856:Chinese
3811:numbers
3739:algebra
3667:Related
3641:Centers
3437:Results
3307:Central
2978:Ptolemy
2973:Proclus
2938:Perseus
2893:Marinus
2873:Hypatia
2863:Hippias
2838:Geminus
2828:Eudoxus
2818:Eudemus
2788:Diocles
2553:. MAA.
878:similar
767:History
508:
484:
466:is the
277:section
97:on the
3871:Indian
3648:Cyrene
3180:Optics
3099:Conics
3018:Theano
3008:Thales
3003:Sporus
2948:Philon
2933:Pappus
2823:Euclid
2753:Carpus
2743:Bryson
2611:
2557:
2530:
2373:
773:Euclid
362:, the
292:vertex
85:, the
3866:Incan
3787:logic
3563:Other
3331:Chord
3324:Axiom
3302:Angle
2958:Plato
2848:Heron
2768:Conon
783:Proof
391:is a
358:is a
294:with
101:in a
3932:Area
3828:list
3116:Data
2888:Leon
2738:Bion
2609:ISBN
2555:ISBN
2528:ISBN
2479:and
2447:and
1993:and
1370:>
876:are
616:and
530:The
352:and
320:and
286:and
273:the
231:The
120:and
3730:by
3444:In
2635:at
1997:DBC
1990:ADC
1983:ABC
1867:cot
1855:tan
1814:tan
1802:tan
1578:BDC
1576:, △
1574:ADC
1400:ABC
1205:ABC
1198:BCD
1196:, △
1194:ACD
1046:BCD
1044:, △
1042:ABC
892:ACD
890:, △
888:ABC
873:BCD
871:, △
869:ADC
520:is
89:or
81:In
3928::
2577:.
2506:^
2465:pq
2452:+
2442:+
1986:,
1765:90
1727:90
1671:90
1088:90
934:90
862::
763:.
625:+
592:.
581:DC
523:BF
517:DC
470:.
463:DC
457:DC
355:CB
349:AC
329:+
309:AE
223:.
2665:e
2658:t
2651:v
2617:.
2587:.
2563:.
2536:.
2481:q
2477:p
2460:h
2454:h
2450:q
2444:h
2440:p
2435:h
2399:.
2396:q
2393:p
2390:=
2381:2
2377:h
2363:q
2360:p
2357:2
2354:=
2342:2
2338:q
2329:2
2325:p
2316:2
2312:)
2308:q
2305:+
2302:p
2299:(
2296:=
2284:2
2280:q
2271:2
2267:p
2258:2
2254:c
2250:=
2238:2
2234:q
2225:2
2221:p
2212:2
2208:b
2204:+
2199:2
2195:a
2191:=
2182:2
2178:h
2174:2
2138:2
2134:b
2130:+
2125:2
2121:a
2117:=
2108:2
2104:c
2094:2
2090:p
2081:2
2077:b
2073:=
2064:2
2060:h
2050:2
2046:q
2037:2
2033:a
2029:=
2020:2
2016:h
1995:△
1988:△
1981:△
1950:q
1947:p
1942:=
1939:h
1928:q
1925:p
1919:2
1915:h
1909:=
1906:1
1895:q
1892:p
1886:2
1882:h
1876:=
1844:q
1841:h
1831:p
1828:h
1823:=
1761:=
1751:)
1748:D
1745:C
1742:A
1723:(
1720:+
1717:D
1714:C
1711:A
1705:=
1695:)
1692:C
1689:B
1686:D
1667:(
1664:+
1661:D
1658:C
1655:A
1649:=
1639:B
1636:C
1633:D
1627:+
1624:D
1621:C
1618:A
1612:=
1605:B
1602:C
1599:A
1572:△
1558:B
1555:D
1552:C
1546:=
1543:C
1540:D
1537:A
1514:.
1508:h
1505:q
1499:=
1493:p
1490:h
1467:q
1464:p
1461:=
1456:2
1452:h
1441:C
1427:q
1424:p
1421:=
1416:2
1412:h
1398:△
1376:)
1373:0
1367:q
1364:,
1361:p
1358:,
1355:h
1352:(
1346:q
1343:p
1338:=
1335:h
1327:q
1324:p
1321:=
1316:2
1312:h
1301:h
1298:q
1293:=
1288:p
1285:h
1258:.
1255:D
1252:C
1249:B
1240:C
1237:B
1234:A
1225:D
1222:C
1219:A
1203:△
1192:△
1175:.
1172:D
1169:C
1166:B
1157:C
1154:B
1151:A
1128:;
1125:D
1122:B
1119:C
1113:=
1110:C
1107:B
1104:A
1097:,
1084:=
1081:C
1078:D
1075:B
1069:=
1066:B
1063:C
1060:A
1040:△
1024:.
1021:D
1018:C
1015:A
1006:C
1003:B
1000:A
974:;
971:D
968:A
965:C
959:=
956:C
953:A
950:B
943:,
930:=
927:C
924:D
921:A
915:=
912:B
909:C
906:A
886:△
867:△
843:C
840:B
837:D
828:C
825:D
822:A
813:C
810:B
807:A
736:q
733:p
730:=
725:2
721:h
713:|
709:B
706:D
702:|
697:|
693:D
690:A
686:|
682:=
678:|
674:E
671:D
667:|
662:|
658:D
655:C
651:|
641::
627:q
623:p
618:q
614:p
590:p
586:h
584:(
564:1
558:p
553:=
550:h
540:q
512:q
494:p
468:h
442:q
439:p
419:q
416:p
413:=
408:2
404:h
379:C
376:B
373:A
340:C
336:D
331:q
327:p
322:B
318:A
314:D
304:p
300:q
296:D
288:q
284:p
255:q
252:p
249:=
244:2
240:h
202:.
190:q
187:p
184:=
179:2
175:h
146:q
143:p
138:=
135:h
122:q
118:p
114:h
64:q
61:p
56:=
53:h
47:q
44:p
41:=
36:2
32:h
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