Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.
Let ABCD be a square of side a.
Therefore its diagonal
Two desired equilateral triangles are formed as ABE and DBF.
Length of one side of ABE = a
Length of one side of DBF
We know that equilateral triangles have all angles as 60o. So, all equilateral triangles are similar to each other.
So, ratio between areas of these triangles will be equal to the square of the ratio of their corresponding sides of these triangles.
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