4. Fixed Point Game

Consider a game plan that is formed by an arbitrary triangulation inside a square. An illustration of one such plan can be found in the figure below. On this plan, a two-player game of player $G$ and $R$ is played. At the beginning of the game, the player $G$ occupy the vertices $a$ and $c$, while the player $R$ occupy the vertices $b$ and $d$. Players alternate in turns, and in each turn a player may occupy (paint with his color) any previously unoccupied vertex. A player wins if he manages to occupy all the vertices on some path between his initial vertices. If the player who is currently on the turn has nowhere else to move, but neither player has completed their path, the game ends in a draw. The starting player is the player $G$.

Your task is to prove the following claim: