The Triangle of Abū’l-Wafā’

I found an interesting geometric statement in a paper of Glen Van Brummelen cited in the online MAA January 2020 issue of Convergence:

“For instance, Abū’l-Wafā’ describes how to embed an equilateral triangle in a square, as follows: extend the base GD by an equal distance to E. Draw a quarter circle with centre G and radius GB; draw a half circle with centre D and radius DE. The two arcs cross at Z. Then draw an arc with centre E and radius EZ downward, to H. If you draw AT = GH and connect B, H, and T, you will have formed the equilateral triangle.”

So the challenge is to prove this statement regarding yet another fascinating appearance of an equilateral triangle.

See the Triangle of Abū’l-Wafā’

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