Problems

In the acute-angled triangle $ABC$, the heights $A{A}_1$ and $B{B}_1$ are drawn. Prove that $A_{1}C\times BC=B_{1}C\times AC$.

Let $A{A}_1$ and $B{B}_1$ be the heights of the triangle $ABC$. Prove that the triangles $A_1{B}_{1}C$ and $ABC$ are similar.