Question 182
Let $ABC$ be a triangle, and let $I$ be the incentre of $ABC$. Let $A'$, $B'$ and $C'$ be three points lying on $(AI)$, $(BI)$ and $(CI)$ respectively. Suppose that $A'$, $B'$ and $C'$ are distinct from $A$, $B$, $C$ and $I$, and that they are collinear. Finally, let $P_A$ be the intersection point of the perpendicular bisectors of $[BB']$ and $[CC']$. Similarly, let $P_B$ be the intersection point of the perpendicular bisectors of $[CC']$ and $[AA']$, and let $P_C$ be the intersection point of the perpendicular bisectors of $[AA']$ and $[BB']$.
Prove that the circumcircles of the triangles $ABC$ and $P_A P_B P_C$ are tangent to each other.
Prove that the circumcircles of the triangles $ABC$ and $P_A P_B P_C$ are tangent to each other.