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side and if the two sides are the legs of a right angle the parallelogram over the third side will be square as well. For a right-angled triangle, two parallelograms attached to the legs of the right angle yield a rectangle of equal area on the third side and again if the two parallelograms are squares then the rectangle on the third side will be a square as well.
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the two arbitrary parallelograms attached to the triangle sides AB and AC. The extended parallelogram sides DE and FG intersect at H. The line segment AH now "becomes" the side of the third parallelogram BCML attached to the triangle side BC, i.e., one constructs line segments BL and CM over BC, such
163:
The theorem generalizes the
Pythagorean theorem twofold. Firstly it works for arbitrary triangles rather than only for right angled ones and secondly it uses parallelograms rather than squares. For squares on two sides of an arbitrary triangle it yields a parallelogram of equal area over the third
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Given an arbitrary triangle with two arbitrary parallelograms attached to two of its sides the theorem tells how to construct a parallelogram over the third side, such that the area of the third parallelogram equals the sum of the areas of the other two parallelograms.
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408:{\displaystyle {\begin{aligned}{\text{A}}_{ABDE}+{\text{A}}_{ACFG}&={\text{A}}_{ABUH}+{\text{A}}_{ACVH}\\&={\text{A}}_{BLQR}+{\text{A}}_{RCMQ}\\&={\text{A}}_{BCML}\end{aligned}}}
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that BL and CM are a parallel and equal in length to AH. The following identity then holds for the areas (denoted by A) of the parallelograms:
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Relates areas of three parallelograms attached to three sides of an arbitrary triangle
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38:. The theorem, which can also be thought of as a generalization of the
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have the same area, the same argument applying to the parallelograms
153:{\displaystyle {\text{A}}_{ABDE}+{\text{A}}_{ACFG}={\text{A}}_{BCML}}
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Due to having the same base length and height the parallelograms
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204:. This already yields the desired result, as we have:
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describes the relationship between the areas of three
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Charming Proofs: A Journey Into
Elegant Mathematics
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465:The Pythagorean Theorem: A 4,000-year History
439:. Mathematical Association of America, 1983,
426:Pappus's Extension of the Pythagorean Theorem
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437:Great Moments in Mathematics (before 1650)
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1316:Straightedge and compass construction
46:(4th century AD), who discovered it.
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1281:Incircle and excircles of a triangle
467:. Princeton University Press, 2007,
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488:Claudi Alsina, Roger B. Nelsen:
23:dark grey area = light grey area
1618:A History of Greek Mathematics
1131:The Quadrature of the Parabola
62:be the arbitrary triangle and
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1399:Intersecting secants theorem
1394:Intersecting chords theorem
1261:Doctrine of proportionality
1856:
1090:On the Sphere and Cylinder
1043:On the Sizes and Distances
1792:Ancient Greece portal
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1596:Philosophy of mathematics
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1511:Ptolemy's table of chords
566:Ancient Greek mathematics
1840:Theorems about triangles
1835:Euclidean plane geometry
1463:Aristarchus's inequality
1036:On Conoids and Spheroids
1571:Ancient Greek astronomy
1384:Inscribed angle theorem
1374:Greek geometric algebra
1029:Measurement of a Circle
522:The Pappus Area Theorem
1805:Mathematics portal
1591:Non-Euclidean geometry
1546:Mouseion of Alexandria
1419:Tangent-secant theorem
1369:Geometric mean theorem
1354:Exterior angle theorem
1349:Angle bisector theorem
1053:On Sizes and Distances
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1429:Theorem of the gnomon
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1177:Problem of Apollonius
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1246:Circumscribed circle
1063:On the Moving Sphere
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1601:Neusis construction
1521:Spiral of Theodorus
1414:Pythagorean theorem
1359:Euclidean algorithm
1301:Lune of Hippocrates
1170:Squaring the circle
926:Theon of Alexandria
601:Aristaeus the Elder
40:Pythagorean theorem
1488:Menelaus's theorem
1478:Irrational numbers
1291:Parallel postulate
1266:Euclidean geometry
1234:Apollonian circles
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65:
61:
56:
49:
47:
45:
41:
37:
33:
29:
21:
1796:
1783:
1625:Thomas Heath
1616:
1499:
1492:
1483:Law of sines
1339:
1271:Golden ratio
1136:
1129:
1120:
1114:(Theodosius)
1110:
1102:
1095:
1088:
1081:
1072:
1062:
1056:(Hipparchus)
1052:
1042:
1034:
1027:
1018:
1008:
1000:
995:(Apollonius)
991:
983:
971:
946:Zeno of Elea
706:Eratosthenes
696:Dionysodorus
528:
521:
508:Google Books
506:, p. 77, at
501:
489:
483:Google Books
481:, p. 58, at
476:
464:
455:Google Books
453:, p. 37, at
448:
436:
425:
201:
197:
193:
189:
185:
181:
177:
173:
171:
162:
67:
63:
59:
57:
53:
27:
26:
1692:mathematics
1500:Arithmetica
1097:Ostomachion
1066:(Autolycus)
985:Arithmetica
761:Hippocrates
691:Dinostratus
676:Dicaearchus
606:Aristarchus
1819:Categories
1744:Babylonian
1644:arithmetic
1610:History of
1439:Apollonius
1124:(Menelaus)
1083:On Spirals
1002:Catoptrics
941:Xenocrates
936:Thymaridas
921:Theodosius
906:Theaetetus
886:Simplicius
876:Pythagoras
861:Posidonius
846:Philonides
806:Nicomachus
801:Metrodorus
791:Menaechmus
746:Hipparchus
736:Heliodorus
686:Diophantus
671:Democritus
651:Chrysippus
621:Archimedes
616:Apollonius
586:Anaxagoras
578:(timeline)
419:References
1830:Equations
1205:Inscribed
965:Treatises
956:Zenodorus
916:Theodorus
891:Sosigenes
836:Philolaus
821:Oenopides
816:Nicoteles
811:Nicomedes
771:Hypsicles
666:Ctesibius
656:Cleomedes
641:Callippus
626:Autolycus
611:Aristotle
591:Anthemius
1769:Japanese
1754:Egyptian
1697:timeline
1685:timeline
1673:timeline
1668:geometry
1661:timeline
1656:calculus
1649:timeline
1637:timeline
1340:Elements
1186:Concepts
1148:Problems
1121:Spherics
1111:Spherics
1076:(Euclid)
1022:(Euclid)
1019:Elements
1012:(Euclid)
973:Almagest
881:Serenus
856:Porphyry
796:Menelaus
751:Hippasus
726:Eutocius
701:Domninus
596:Archytas
461:Eli Maor
36:triangle
1749:Chinese
1704:numbers
1632:algebra
1560:Related
1534:Centers
1330:Results
1200:Central
871:Ptolemy
866:Proclus
831:Perseus
786:Marinus
766:Hypatia
756:Hippias
731:Geminus
721:Eudoxus
711:Eudemus
681:Diocles
503:excerpt
478:excerpt
450:excerpt
50:Theorem
1764:Indian
1541:Cyrene
1073:Optics
992:Conics
911:Theano
901:Thales
896:Sporus
841:Philon
826:Pappus
716:Euclid
646:Carpus
636:Bryson
496:
471:
443:
1759:Incan
1680:logic
1456:Other
1224:Chord
1217:Axiom
1195:Angle
851:Plato
741:Heron
661:Conon
430:JSTOR
168:Proof
1825:Area
1721:list
1009:Data
781:Leon
631:Bion
494:ISBN
469:ISBN
441:ISBN
202:RCMQ
200:and
198:ACVH
194:BLQR
192:and
190:ABUH
186:ACVH
184:and
182:ACFG
178:ABUH
176:and
174:ABDE
68:ACFG
66:and
64:ABDE
58:Let
1623:by
1337:In
60:ABC
1821::
463::
196:,
188:,
558:e
551:t
544:v
510:)
485:)
457:)
432:)
397:L
394:M
391:C
388:B
383:A
378:=
366:Q
363:M
360:C
357:R
352:A
347:+
342:R
339:Q
336:L
333:B
328:A
323:=
311:H
308:V
305:C
302:A
297:A
292:+
287:H
284:U
281:B
278:A
273:A
268:=
259:G
256:F
253:C
250:A
245:A
240:+
235:E
232:D
229:B
226:A
221:A
146:L
143:M
140:C
137:B
132:A
127:=
122:G
119:F
116:C
113:A
108:A
103:+
98:E
95:D
92:B
89:A
84:A
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