Students will learn what both variants of the geometric mean theorem are, why they're true, and how to use them. Before attempting the proofs, students must know the AA similarity theorem, which is a direct consequence of the third angle theorem. Here's a proof of the altitude theorem, and here's a proof of the leg theorem. Here's a video showing some problems which can be easily solved using the altitude theorem, and here's one for the leg theorem. Note that the two aforementioned videos are a good source of problems, but the method used is incorrect, as the instructor does not apply the theorems directly.