A triangle $ABC$ is given and two points $P_1$ and $P_2$ inside it.
We know that: $AP_1\cap BC=A_1$, $BP_1\cap AC=B_1$, $CP_1\cap AB=C_1$, $AP_2\cap BC=A_2$, $BP_2\cap AC=B_2$, $CP_2\cap AB=C_2$. We also know that: $A_1B_1\cap B_2C_2=N$, $A_1C_1\cap B_2C_2=P$ and $A_1C_1\cap A_2B_2=M$.
Show that $AM,BN$ and $CP$ are concurrent.
I think we can use Ceva's Theorem for the concurrence of cevians in a triangle. Can you help me?