How to get as much pie as possible! Alice and Bob are sharing a triangular pie. Alice will cut the pie with one straight cut and pick the bigger piece, but Bob first specifies one point through which the cut must pass. What point should Bob specify to get as much pie as possible? And in that case how much pie can Alice get?
The approach I took was to put an arbitrary point in a triangle, and draw lines from it to each corner of the triangle. Now the cut will only go through two of the three triangles we now have, so the aim is to get as close to 50% of the remaining two triangles as possible?

If the triangle is equilateral, it seems like the best answer would be for Bob to pick the center of the triangle, in which case Alice cuts one of the corners off and leaves Bob with 4/9 of the cake.
The reason that any point other than the center will not work is because Alice can simply cut parallel to the furthest side from the point, and will leave Bob with less than 4/9.
And if the problem is affine-invariant like Christian Blatter claims it to be, then the point on non-equilateral triangles should be a specific center called the centroid, the intersection of the three medians. And the percentage Bob can expect to get is still 4/9 as usual.
