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)
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.