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

Explanation

GIVEN A `Delta ABC` in which `angle ABC=90^(@) and AB= BC. Delta ABD and Delta CAE` are equilateral triangles.
TO PROOF `ar (Delta ABD) =(1)/(2)xx ar (Delta CAE)`
PROOF Let `AB=BC=x` units.

`:. hyp . CA= sqrt(x^(2)+x^(2)) = xsqrt(2)`. units
Each of the `Delta ABD and CAE` being equilateral, each angle of each one of them is `60^(@)`
`:. Delta ABD~ Delta CAE`[ by AAA- similarity]
But, the ratio of the areas of two similar triangles is equal to the ratio of the sequare sof their corresponding sides.
`:. (ar (Delta ABD))/(ar (Delta CAE))=(AB^(2))/(CA^(2))=(x^(2))/((xsqrt(2))^(2)) =(x^(2))/(2x^(2))=(1)/(2)`
Hence, `ar (Delta ABD)=(1)/(2)xx ar (Delta CAE)`