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Explanation of the construction: We need a length s such that s^2 = AB*BC. This can be written s/AB = BC/s, which suggests similar triangles. The triangle AGE has a right angle at G, being inscribed in a semicircle, and triangles ABG and GBE are right triangles similar to it and thus each other. We have thus equal ratios of short leg to long leg, BG/AB = BE/BG, and since BE = BC, we can use BG for the side s.
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