Let O designate the centre of an equilateral triangle. Points U-Z are chosen at random within the triangle. We have learnt that points U,V,W are each nearer to a (possibly different) vertex than to O; while X,Y are each closer to O than to any of the vertices.

Show that triangle XYZ is more likely than triangle UVW to contain the point O within its interior.

Very satisfying explanation.

Up through your b) (inclusive) our approaches were identical. Then in c) you exposed a fact which had eluded me.

If you are still interested, you can try to work out the probability of enclosing O when exactly one of the points X,Y,Z lie inside the hexagon. I don't know the answer myself, so if you solve it, I'd like to hear from you.