Huygens' Proof of the Pythagorean Theorem

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A few words and an explanation.

Explanation

The famous Dutch mathematician, astronomer and physicist Christiaan Huygens (1629 � 1695) published in 1657 a proof of the Pythagorean theorem introducing it as "Demonstratio mea Propos^is. 47 lib. I Euclidis ". It was included in Loomis' collection as geometric proof #31 and in 1936 brought up in a dedicated note in The Mathematical Gazette.

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Start with the Bride's Chair diagram. Extend GA to P such that AP = AG and extend BD to N so that BN = BD. Join A with N and P with K. Let LM be parallel to AB through P. As in Proof #69, the main instrument in the proof is Euclid's I.41: if a parallelogram and a triangle that share the same base and are in the same parallels (I.41), the area of the parallelogram is twice that of the triangle.

The area of ΔABP with base AB and altitude from equal to AC is half that of the square on side AC. The area of ΔANB is half the area of the square on side BC. In ΔAPS,

AP = AD = AC,
AS = AB,
∠PAS = ∠BAC (as two have perpendicular sides)

implying ΔAPS = ΔABC. In particular, PS = BC = BN. Thus in ΔKPS,

PS = BN,
KS = AB,
∠KPS = 90° - ∠ASP = 90° - ∠ABC = ∠ABN

implying ΔKPS = ΔANB and making the area of ΔKPS half that of the square on BC. On the other hand, the area of ΔABP is half the area of rectangle ABML while the area of ΔKPS is half that of rectangle KMLS wherefrom the area of the square on AB is indeed the sum of the areas of the squares on AC and BC.

References

E. S. Loomis, The Pythagorean Proposition, NCTM, 1968
J. P. McCarthy, Huygens' Proof of the Theorem of Pythagoras, The Mathematical Gazette, Vol. 20, No. 240. (Oct., 1936), pp. 280-281.

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