CGMO 2013 Problem 1

Let $A$ be the closed region bounded by the following three lines in the $xy$ plane: $x=1, y=0$ and $y=t(2x-t)$ , where $0<t<1$ . Prove that the area of any triangle inside the region $A$ , with two vertices $P(t,t^2)$ and $Q(1,0)$ , does not exceed $\frac{1}{4}.$