KöMaL 2012 A570

Given a triangle ABC . For an arbitrary interior point $X$ of the triangle denote by $A_1$($X$) the point intersection of the lines $AX$ and $BC$, denote by $B_1$($X$) the point intersection of the lines $BX$ and $CA$, and denote by $C_1$($X$) the point intersection of the lines $CX$ and $AB$. Construct such a point $P$ in the interior of the triangle for which each of the quadrilaterals $AC_1$($P$)$PB_1$($P$), $BA_1$($P$)$PC_1$($P$) and $CB_1$($P$)$PA_1$($P$) has an inscribed circle.