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

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.