Question 5
Let $P$ be a point inside $\triangle ABC$. Let $A_1, B_1, C_1$ be points in the interiors of the segments $PA, PB, PC$, respectively. Let $\overline{BC_1} \cap \overline{CB_1} = \{A_2\}$, $\overline{CA_1} \cap \overline{AC_1} = \{B_2\}$, and $\overline{AB_1} \cap \overline{BA_1} = \{C_2\}$. Let $U$ be the intersection of the lines $A_1B_1$ and $A_2B_2$, and $V$ be the intersection of the lines $A_1C_1$ and $A_2C_2$. Show that the lines $UC_2$, $VB_2$, and $AP$ are concurrent.