Show that $P$ lies inside an even number of triangles with vertices among $A_1 , . . . , A _{2m}$

Question

Assume a convex $2m$-gon $A_1 , . . . , A _{2m}$ . In its interior we choose a point $P$ , which does not lie on any diagonal. Show that $P$ lies inside an even number of triangles with vertices among $A_1 , . . . , A _{2m}.$

This is from Problem solving strategies, Invariance chapter. I tried to find the invariance, but couldn't get one till now.

Any hints would be helpful.

Answer 1

Let the point be outside. In this case, it lies inside even number of triangles i.e., zero. Now move that point through every region in such a way its locus does not cross any intersection point. Suppose that point crosses $A_iA_j$ then it goes out of all triangles of form $A_iA_jA_k$ and moves inside all triangles of form $A_iA_jA_m$ where $k$ and $m$ are on opposite sides of $A_iA_j$. As the polygon has even number of vertices, parity of number of triangles on each side is same, so no change in parity of number of triangles that have that point inside them. As initially it was even, it will always remain even.