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Question

Prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangle described on its diagonal.

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Solution

Given: A square ABCD,
An equilateral BCE described on side BC of the square.
An equilateral BDF described on the diagonal BD of the square.
BCEBDF ( both are equiangular, each angle=60)
ar(BCE)ar(BDF)=BC2BD2=BC2(2BC)2 ( diagonal of a square=2 side)
=BC22BC2=12.

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