Question 117
Let $D$ be a point on side $AB$ of triangle $ABC$ such that $CD = AC$, as shown in the figure below. The incircle of triangle $ABC$ is tangent to the sides $AC$ and $AB$ at the points $E$ and $F$, respectively. Let $I$ be the incenter of triangle $BCD$ and $P$ the intersection point of the segments $AI$ and $EF$. Moreover, let $G$ be a point on the segment $AB$ such that $IG$ and $EF$ are parallel.
a) Prove that $DI = IG$.
b) Prove that $AP = PI$.
Remark: The incenter of a triangle is the intersection point of the bisectors of the interior angles of the triangle, and the incircle is the circle centered at the incenter and tangent to the three sides of the triangle.
a) Prove that $DI = IG$.
b) Prove that $AP = PI$.
Remark: The incenter of a triangle is the intersection point of the bisectors of the interior angles of the triangle, and the incircle is the circle centered at the incenter and tangent to the three sides of the triangle.