KöMaL 2008 B4067

Erect a perpendicular to each of the sides $AB$, $BC$, $CA$ of a triangle of perimeter $k$ at the vertices $A$, $B$, $C$, respectively. Let $K$ denote the perimeter of the new triangle obtained from the perpendiculars. Prove that $\frac{K}{k}= \cot\alpha+ \cot\beta+ \cot\gamma$, where $\alpha$, $\beta$ and $\gamma$ are the angles of the triangle ABC .