In triangle $ABC$, $A'$ is the midpoint of $BC$, $B'$ is the midpoint of $AC$, and the line segments $AA'$ and $BB'$ are perpendicular. Let $AC = b$, $BC = a$, and $AB = c$.
- Find $c$ in terms of $a$ and $b$.
- Determine the values of $b$ in terms of $a$ for which the triangle exists.
- Calculate the size of the maximum interior angle at vertex $C$.
Problem ID: 296 (10 Dec 2006) Difficulty: 4 Star