Question 23
The inscribed circle of the triangle $A_{1} A_{2} A_{3}$ touches the sides $A_{2} A_{3}$, $A_{3} A_{1}$ and $A_{1} A_{2}$ at points $S_{1}$, $S_{2}$, $S_{3}$, respectively. Let $O_{1}$, $O_{2}$, $O_{3}$ be the centres of the inscribed circles of triangles $A_{1} S_{2} S_{3}$, $A_{2} S_{3} S_{1}$ and $A_{3} S_{1} S_{2}$, respectively. Prove that the straight lines $O_{1} S_{1}$, $O_{2} S_{2}$ and $O_{3} S_{3}$ intersect at one point.