1.Let $\triangle ABC$ be a right triangle, with the point $H$ the foot of the altitude from $C$ to side $\overline{AB}$.
Prove that
(x+ h)^2 + (y + h)^2 = (a + b)^2 expand
x^2 + 2xh + h^2 + y^2 + 2yh + h^2 = a^2 + 2ab + b^2
By the Pythagorean Theorem.....
(x^2 + h^2) + ( y^2 + h^2) = a^2 + b^2
So....we have left that
2xh + 2yh = 2ab
xh + yh = ab
h (x + y) = ab multiply both sides by 1/2
(1/2) (x + y) * h = (1/2)ab
Since this is a right triangle (1/2) product of legs a, b = the area of ABC = the right side
And the left side is (1/2)base of ABC * height = the area of ABC
So...the left side = the right side