Question 42
Consider a triangle $\triangle ABC$.
a) Prove that the interior bisector of the angle $\angle A$ and the exterior bisectors of the angles $\angle B$ and $\angle C$ intersect at a point $I_A$.
b) Let $I_A M \perp AC$, $M \in AC$, $I_A N \perp BC$, $N \in BC$ and $I_A P \perp AB$, $P \in AB$. Show that if $\overrightarrow{I_A M} + \overrightarrow{I_A P} = \overrightarrow{I_A N}$, then the triangle $ABC$ is equilateral.
a) Prove that the interior bisector of the angle $\angle A$ and the exterior bisectors of the angles $\angle B$ and $\angle C$ intersect at a point $I_A$.
b) Let $I_A M \perp AC$, $M \in AC$, $I_A N \perp BC$, $N \in BC$ and $I_A P \perp AB$, $P \in AB$. Show that if $\overrightarrow{I_A M} + \overrightarrow{I_A P} = \overrightarrow{I_A N}$, then the triangle $ABC$ is equilateral.