 Multiplying through by yields

Considering the two other altitudes of the triangle yields

Adding the latter two equations gives

Subtracting the first equation from the last one results in

which simplifies to

This proof uses trigonometry in that it treats the cosines of the various angles as quantities in their own right.It uses the fact that the cosine of an angle expresses the relation between the two sides enclosing that angle in "any" right triangle.Other proofs (below) are more geometric in that they treat an expression such as merely as a label for the length of a certain line segment.