USAMO 1990 Problem 5

$\mathrm{ABC}$ is acute-angled. The circle diameter $\mathrm{AB}$ meets the altitude from $\mathrm{C}$ at $\mathrm{P}$ and $\mathrm{Q}$. The circle diameter $\mathrm{AC}$ meets the altitude from B at R and S. Show that P, Q, R and S lie on a circle.