Heron's Formula

Begininning with a right triangle of sides a, b, and c, (a is the hypotenuse, b is the base, c is the height) determine its area in two ways; by the standard A = (1/2)(base)(height) and by Heron's Formula. Equate these and manipulate the resulting, dreadful equation until you have derived the theorem of Pythagoras; a^2 = b^2 + c^2.

See attached for Heron's Formula
Attachment 19345

What you did so far...?

4(b^2)(c^2) = -1(a^4 + b^4 + c^4) + 2((a^2)(b^2) + (a^2)(c^2) + (b^2)(c^2))