1962 AHSME Problems/Problem 23
In triangle , is the altitude to and is the altitude to . If the lengths of , , and are known, the length of is:
We can actually determine the length of no matter what type of angles and are. This can be easily proved through considering all possible cases, though for the purposes of this solution, we'll show that we can determine if is an obtuse angle.
Let's see what happens when is an obtuse angle. by SSS, so . Hence . Since we've determined the length of even though we have an obtuse angle, is not determined by what type of angle may be. Hence our answer is .