Question 34
Let $I$ be the incenter of triangle $ABC$ and let $k$ be a circle through the points $A$ and $B$. This circle intersects
* the line $AI$ in points $A$ and $P$,
* the line $BI$ in points $B$ and $Q$,
* the line $AC$ in points $A$ and $R$ and
* the line $BC$ in points $B$ and $S$,
with none of the points $A, B, P, Q, R$ and $S$ coinciding and such that $R$ and $S$ are interior points of the line segments $AC$ and $BC$, respectively.
Prove that the lines $PS, QR$ and $CI$ meet in a single point.
(Stephan Wagner)
* the line $AI$ in points $A$ and $P$,
* the line $BI$ in points $B$ and $Q$,
* the line $AC$ in points $A$ and $R$ and
* the line $BC$ in points $B$ and $S$,
with none of the points $A, B, P, Q, R$ and $S$ coinciding and such that $R$ and $S$ are interior points of the line segments $AC$ and $BC$, respectively.
Prove that the lines $PS, QR$ and $CI$ meet in a single point.
(Stephan Wagner)