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?

You can consider the triangle to be equilateral, as you can make any triangle equilateral with a linear transformation.  That transformation will preserve the ratio of areas.  The centroid is the obvious point to pick.  If Alice cuts parallel to a side, she leaves Bob $\frac 49$ because the centroid is $\frac 23$ of the way along the altitude.
