Cathetus theorem/Linear algebra/Fact

Cathetus theorem

Let ${}A,B,C$ be a right triangle, with the right angle located at ${}C$. Let ${}h$ denote the altitude through ${}C$, and let ${}D$ denote the foot of this altitude on the line given by ${}A$ and ${}B$.

Then

${}d(A,C)^{2}=d(A,D)\cdot d(A,B)\,.$

Proof 1, 2, Write another proof