Oxford University MAT 2019 Solved Papers

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