Area of a triangle is obviously rp, where r is the inradius and p = (a + b + c)/2 the semiperimeter of the triangle. From the diagram, the hypothenuse c = (a - r) + (b - r), or r = p - c. The area of the triangle then is computed in two ways:
p(p - c) = ab/2,
which is equivalent to
(a + b + c)(a + b - c) = 2ab,
or
(a + b)² - c² = 2ab.
And finally
a² + b² - c² = 0.