## Task 4 - 050834

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0604.jpg?height=366&width=551&top_left_y=314&top_left_x=273)

The incircle $k$ of triangle $ABC$ touches the sides $AB$, $BC$, and $CA$ at points $D$, $E$, and $F$ (see figure). The angles of triangle $ABC$ have measures $\alpha$, $\beta$, and $\gamma$.

Determine the measures of the angles $\angle DEF$, $\angle EFD$, and $\angle FDE$ of triangle $DEF$!
If the center of the incircle is denoted by $M$, then $\triangle C F M \cong \triangle C M E$, as they coincide in 2 sides and the right angle.

Consequently, $\overline{C F}=\overline{C E}$ and thus $C M \perp E F$,

and due to (1) and $M E \perp E G$ also $\angle M E F \cong \angle M F E \cong \angle M C E$.

Similarly, we obtain $\angle M E D \cong \angle M D E \cong \angle M B E$

and $\angle M D F \cong \angle M F D \cong \angle M A F$.

The measure of the angle $\angle M C E$ is $\frac{\gamma}{2}$, that of the angle $\angle M B E$ is $\frac{\beta}{2}$, and that of the angle $\angle M A F$ is $\frac{\alpha}{2}$, since $C M$ or $B M$ or $A M$ are the angle bisectors of the triangle $\triangle A B C$. If the measure of the angle $\angle D E F$ is denoted by $\delta$, that of the angle $\angle E F D$ by $\epsilon$, and that of the angle $\angle F D E$ by $\eta$, then due to (2) and (3) $\delta=\frac{\gamma+\beta}{2}$, due to (2) and (4) $\epsilon=\frac{\gamma+\alpha}{2}$, and due to (3) and (4) $\eta=\frac{\alpha+\beta}{2}$.