3-colourability of Eulerian maximal planar graph
When you make deductions in this coloring problem you are following paths in the dual graph to the triangulation. Any inconsistency could be described by a cycle in the dual graph (a cycle of triangles linked edge-to-edge in the given maximal planar graph) such that, when you color one of the triangles (it doesn't matter which one or which coloring) and then propagate information around the cycle you come back to something different.
If you have a dual cycle that surrounds only a single primal vertex, then it's a wheel, and it's inconsistent if and only if it's an odd wheel.
If you have a dual cycle that surrounds more than one primal vertex, you can draw a dual path inside it that splits it into two dual cycles that each surround fewer primal vertices (this is where we use the fact that the triangulation is on the plane and not on a higher-genus surface). If both smaller cycles are consistent you could glue together consistent colorings of both to get a consistent coloring of the whole thing. So by induction on number of primal vertices contained, when the graph is Eulerian, all dual cycles are consistent.