Question 83
Acute triangle $A B C$ is made of solid metal, and it is on top of a wooden table. Points $P$ on $A B$ and $Q$ on $A C$ are such that the perpendicular to $A B$ through $P$ intersects the perpendicular to $A C$ through $Q$ inside the triangle. Nails are hammered into the table at $P$ and $Q$. (The nails do not go through the triangle but are at its edges, bounding the triangle.) Prove that there is a unique position $R$ on $B C$ such that, if a nail is hammered into the table at $R$, the triangle will no longer be able to move (within the plane of the table).