ISL 1979 Problem 62

$T$ is a given triangle with vertices $P_1,P_2,P_3$ . Consider an arbitrary subdivision of $T$ into finitely many subtriangles such that no vertex of a subtriangle lies strictly between two vertices of another subtriangle. To each vertex $V$ of the subtriangles there is assigned a number $n(V)$ according to the following rules: $(\text{i})$ If $V$ = $P_i$ , then $n(V) = i$ . $(\text{ii})$ If $V$ lies on the side $P_i P_j$ of $T$ , then $n(V) = i$ or $j$ . $(\text{iii})$ If $V$ lies inside the triangle $T$ , then $n(V)$ is any of the numbers $1,2,3$ . Prove that there exists at least one subtriangle whose vertices are numbered $1, 2, 3$ .