Prove this Geometry Problem

By the Pythagorean Theorem, we know that  x² + h²  =  a²   and   y² + h²  =  b²

which, if we add them together, we get:    x² + h²  +  y² + h²  =  a² + b²

However, we want to show that (x + h)² + (y + h)²  =  (a + b)²

and, this is:      ( x² + 2xh + h² ) + ( y² + 2yh + h² ) =  a² + 2ab + b²

So, somehow we will need to show tha 2xh + 2yh  =  2ab

We can do this by finding the area of the triangle in two different ways:

    Area  =  ½ab   and the area is also   Area  =  ½(x +y)h

So:  ½(x +y)h  =  ½ab

--->     (x + y)h  =  ab

--->     xh + yh  =  ab

--->   2xh + 2yh  =  2ab

Adding this to:  x²           + h²  +  y²           + h²  =  a²          + b²

we get:             x² + 2xh + h²  +  y² + 2yh + h²  =  a² + 2ab + b²

which is:              (x + h)²        +     (y + h)²        =      (a + b)²

Let's use real numbers.

a = 3      b = 4      c = 5      h = 2.4       x = 1.8       y = 3.2

(x + h)² + (y + h)² = (a + b)²

(1.8+2.4) + (3.2+2.4)  = (3 + 4)²

17.64 + 31.36 = 49

49 = 49