Pigeonhole and Extremal Arguments · Hard problem

Points $A = (0, 0)$ and $B = (6, 3)$ are given, together with $P = (1, 3)$, $Q = (4, 1)$, $R = (5, 4)$, $S = (6, 0)$, $T = (0, 2)$. The empty-triangle proof chooses $C$ to be the point closest to the LINE $AB$. Which point is $C$?