Question 113
Point $P$ lies inside triangle $ABC$. Let $I_A$, $I_B$, $I_C$ be incenters of triangles $PBC$, $PAC$, $PAB$ respectively. Let $I_P$ denote the incenter of triangle $I_A I_B I_C$. Prove that for point $P$ which satisfies the condition $I_P = P$, the following equalities hold:
$$
AP - BP = AC - BC, \quad BP - CP = BA - CA, \quad CP - AP = CB - AB.
$$
$$
AP - BP = AC - BC, \quad BP - CP = BA - CA, \quad CP - AP = CB - AB.
$$