We study a combinatorial game called Bichromatic Triangle Game, defined as follows. Two players R and B construct a triangulation on a given planar point set V. Starting from no edges, they take turns drawing one straight edge that connects two points in V and does not cross any of the previously drawn edges. Player R uses color red and player B uses color blue. The first player who completes one empty monochromatic triangle is the winner. We show that each of the players can force a tie in the Bichromatic Triangle Game when the points of V are in convex position, and also in the case when there is exactly one inner point in the set V. As a consequence of those results, we obtain that the outcome of the Bichromatic Complete Triangulation Game (a modification of the Bichromatic Triangle Game) is also a tie for the same two cases regarding the set V.
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