I am comfortable using and proving Hero's (Heron's) formula using the tools of algebra. I find myself awestruck contemplating how someone derived the formula without the tools of algebra.
The Pythagorean Theorem can be cast as proving that the sum of the areas of two shapes is equal to the area of another shape - beautiful and approachable. I do not even understand how to cast Hero's as a pure geometry problem.
My thinking so far: Construct the angle bisectors, and then show that the area of the triangle is equal to the area of a rectangle where one side is the semiperimeter and the other side is the radius of the incircle.
I just do not see how to cast the formula in terms of shapes. Any suggestions?