The Pythagorean Proposition: Algebraic Proofs, Proof #89 (Math Lair)

Eighty-Nine

Fig. 87

With the legs of the rt. tri. ABH as radii describe circumferences, and extend AB to C and F. Draw HC, HD, HE, and HF. From the similar tri's AHF and HDH,
AF : AH = AH : AD
∴ b² = AF × AD.---(1)

From the similar tri's CHB and HEB,

CB : HB = HB : BE. ∴a² = CB × BE.---(2)

(1) + (2) = (3) a² + b² = CB × BE + AF × AD

= (h + b)(h - b) + (h + a)(h - a)

= h² − b² + h² − a²;

∴(4) 2h² = 2a² + 2b². ∴ h² = a² + b².

a. Am. Math. Mo., V. IV, p. 12; also on p. 12 is a proof by Richardson. But it is much more difficult than the above method.