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.

ΔABE is described on the side AB of the square ABCD

ΔDBF is described on the diagonal BD of the square ABCD

Since ΔABE and ΔDBF are equilateral triangles

ΔABE ∼ ΔDBF [each angle in an equilateral triangle is 60 degree]

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Area of ΔABE / Area of ΔDBF = ( AB)² /(DB)²

Area of ΔABE / Area of ΔDBF = (AB)² / (√2AB)²  [diagonal of a square is 2 × side]

Area of ΔABE / Area of ΔDBF = AB² / 2AB²

Area of ΔABE / Area of ΔDBF = 1/2

Area of ΔABE = 1/2 × Area of ΔDBF