Recall:

In any triangle, the altitude to a side is equal to the product of the sine of the angle subtending the altitude and a side from the angle to the vertex of the triangle.

In this picutre, the altitude to side c is b sin A or a sin B

Since in the triangle pictured we have (Setting these equal and rewriting as ratios leads to the demonstration of the Law of Sines)

Now we look for a substitution for sin A in terms of a, b, and c. It is readily (if messy) available from the Law of Cosines

and substitution in the identity

Factor (easier than multiplying it out) to get

Rewrite with common denominators.

Factor

Factor and rearrange

Now where the semiperimeter s is defined by

the four expressions under the radical are 2s, 2(s - a), 2(s - b), and 2(s - c). So

Since

we have