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