[Math Lair] The Pythagorean Proposition: Algebraic Proofs, Proof #89

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Eighty-Nine

[Figure 87]
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.

Algebraic Proofs, Proof #1 | Geometric Proofs, Proof #15
For more information on this work, see The Pythagorean Proposition by Elisha Scott Loomis.