Proofs of Heron's formula and the ½|ad - bc| area method

After class on Thursday, a student asked for proofs of our two methods for getting the area of a triangle.

Here is a link to the Wikipedia page on Heron's formula which has several proofs.

For the triangle, here is a page proving |ad - bc| is the area of a parallelogram, which is done by cutting a parallelogram out of a larger rectangle. Any parallelogram can thought of as two congruent triangles glued together along one corresponding side, so half the area of a parallelogram is the area of one of the triangles.