Incircle and Tangency Proof Let the incircle of triangle $ABC$ be tangent to sides $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ at $D$, $E$, and $F$, respectively. Prove that triangle $DEF$ is acute. ![Diagram]( * * * I can't prove that the triangles are similar, and we are not given any specific angle of side measurements. I'm stuck, any answer is greatly appreciated.

$CDE$ is an isosceles triangle, hence $\widehat{EDC}=\frac{\pi-C}{2}$. In a similar way, $\widehat{DBF}=\frac{\pi-B}{2}$, hence: $$ \widehat{FDE}=\pi-\frac{\pi-B}{2}-\frac{\pi-C}{2}=\frac{B+C}{2}$$ and since $B+C<\pi$, $\widehat{FDE}$ is an acute angle. The same applies to $\widehat{FED}$ and $\widehat{DFE}$.