Geometry Game – The Prisoner's Dilemma

Figure 1: Game of Slices; you get the pink, they get the purple.

You and a buddy start with a rectangular cake, represented by rectangle $A B C D$ in Figure 1. Then you get to pick any point $E$ between $C$ and $D$ , and the cake is sliced with two straight cuts, from $B$ to $E$ and $E$ to $A$ . Then it’s your friend’s turn to pick a point $F$ between $B$ and $C$ , so that the cake will be sliced from $D$ to $F$ and $F$ to $A$ . The four straight slices intersect at points $G$ , $H$ , and $I$ as shown in Figure 1. You receive piece $A G H I$ of the cake, and the other player receives pieces $D E G$ , $C E H F$ , and $B F I$ .

Determine all points $E$ you can choose on side $C D$ that will prevent your buddy from ending up with more cake (in all) than you do.

D20. Geometry Game