Oxford University MAT 2019 Solved Papers

Define co-ordinates with the origin is at $\mathit{O}$, and $\mathit{A}$ on the positive $\mathit{x}$-axis. Let $\mathit{\theta }$ be the angle the line makes with the $\mathit{x}$-axis. The length of $\mathit{P}\mathit{Q}$ is a function of $\mathit{\theta }$, call it $\mathit{f}\left(\mathit{\theta }\right)$. This function is even $\mathit{f}\left(-\mathit{\theta }\right)=\mathit{f}\left(\mathit{\theta }\right)$ because the triangle has reflectional symmetry in the $\mathit{x}$-axis. The function therefore has a stationary value at $\mathit{\theta }=0$. The function is also periodic with period ${60}^{\circ }$. Combining these facts, we see that $\mathit{f}\left(\mathit{\theta }\right)$ must take a stationary value halfway between the stationary points at $\mathit{\theta }={30}^{\circ }$, i.e. parallel to one of the sides. The length of $\mathit{P}\mathit{Q}$ in this configuration is $2∕3$.