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ΔABF has base AF and the altitude from B equal to AC.

Its area therefore equals half that of square on the side AC.

I include it on a separate page with Jim's kind permission.

The proof below is a somewhat shortened version of the original Euclidean proof as it appears in Sir Thomas Heath's translation. This is because, and ∠BAF = ∠BAC ∠CAF = ∠CAB ∠BAE = ∠CAE.

Below is a collection of 118 approaches to proving the theorem.

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