Jim Wilson
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. (Setting these equal and rewriting as ratios leads to the demonstration of the Law of Sines)
In the triangle picturedso
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 -- i.e. the projection of sides a and b onto c.
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
