Exercise 19. A triangle $A B C$ is isosceles at $A$. Suppose that all rectangles whose two vertices belong to $[B C]$ and the other two vertices to $[A B]$ and $[A C]$ have a perimeter equal to 84. Determine the value of $A B^{2}$.
Solution to Exercise 19: Let $M N P Q$ be the rectangle, with $M, N \in [B, C]$. Let $D$ be the midpoint of $[B C]$. Set $x = B M$. We have $M Q = x \frac{A D}{B D}$ and $M N = B C - 2 x$. Since $42 = M Q + M N$, we have $\frac{A D}{B D} = 2$ and $B C = 42$, so $A D = 42$ and $A B^{2} = B D^{2} + A D^{2} = 21^{2} + 42^{2} = 2205$.