Question 56
Let $\ell$ and $m$ be two non-coplanar lines in space, and let $P_{1}$ be a point on $\ell$. Let $P_{2}$ be the point on $m$ closest to $P_{1}$, $P_{3}$ be the point on $\ell$ closest to $P_{2}$, $P_{4}$ be the point on $m$ closest to $P_{3}$, and $P_{5}$ be the point on $\ell$ closest to $P_{4}$. Given that $P_{1}P_{2}=5$, $P_{2}P_{3}=3$, and $P_{3}P_{4}=2$, compute $P_{4}P_{5}$.