Equilateral triangles are drawn on the sides of a right triangle. Show that the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.

Explanation

To solve the problem, we need to show that the area of the equilateral triangle drawn on the hypotenuse of a right triangle is equal to the sum of the areas of the equilateral triangles drawn on the other two sides.

Let's denote the right triangle as \( \triangle ABC \) where \( \angle C \) is the right angle. Let the lengths of the sides be \( AB = c \) (the hypotenuse), \( AC = b \), and \( BC = a \).

### Step 1: Calculate the area of the equilateral triangle on the hypotenuse
The area \( A_1 \) of an equilateral triangle with side length \( c \) is given by the formula:
\[
A_1 = \frac{\sqrt{3}}{4} c^2
...