Sept 1: Understand the question and construct a diagram through the given hints

Here ABCD is a square,

Equilateral \triangle DBF on the diagonal of a square

Equilateral \triangle AEB on the side of the square.

Step 2: Show that the two equilateral triangles are similar

NOTE: In every equilateral triangle's angle = 60 degrees.

Therefore, by AAA similarity criterion ,

Equilateral triangles are similar \triangle DBF ~ \triangle AEB

Step 3: Find the ratio of the areas of the smilar triangle.

THEOREM: theorem: The ratio of the areas of two similar triangles is

equal to the ratio of the squares of their corresponding sides.)

= \frac{area \triangle DBF}{area \triangle AEB} = (\frac{DB}{AB})^2 ........................(1)

Step 3: Recall the relation between the side and diagonal of the square

NOTE: Diagonal of square is \sqrt{2} times of its side

EXAMPLE: From the figure DB = \sqrt{2} AB

Step 4: Plugging the diagonal (DB) value in the equation (1)

EXAMPLE: \frac{area \triangle DBF}{area \triangle AEB} = (\frac{DB}{AB})^2

= \frac{area \triangle DBF}{area \triangle AEB} = (\frac{\sqrt{2}AB}{AB})^2

= \frac{area \triangle DBF}{area \triangle AEB} = \frac{2}{1}

= Area \triangle DBF = 2*area \triangle AEB

Hence proved

∴ Area of equilateral triangle described on one side of square is equal to half the area of the equilateral triangle described on one of its diagonals.