# shorter of shortest paths between two points via a pair of lines

The following exercise is from the book What is Mathematics by Richard Courant and Herbert Robbins:

Given two lines L, M and two points P, Q situated inside the angle formed by the two lines, the path of minimum length from P to L, then to M, and then to Q can be found as follows:

Let Q' be the reflection of Q in M and Q'' the reflection of Q' in L; draw PQ'' intersecting L in R and RQ' intersecting M in S; then R and S are the required points such that PR+RS+SQ is the path of minimum length from P to L to M to Q.
One might ask for the shortest path first from P to M, then to L, and from there to Q. In this case Q' is the reflection of Q in L and Q'' the reflection of Q' in M. R is the intersection of PQ'' with M and S the intersection of RQ' with L.

Show that the first path is smaller than the second if O (where O is the point of intersection between L and M) and R lie on the same side of the line PQ.

How do you do this?

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The diagrams above depicts straight lines obtaining by developing 3D helical geodesics on a cone of semi-vertical angle$\, \pi \sin^{−1}LOM.$ Generators $OL,OM$ are diametrically opposite lines of the cone seen when a thin flexible sheet cone is flattened to a plane,when the geodesic curved helices now appear as straight lines by virtue of Gauss Egregium Theorem (in this particular case straight lines map to straight lines). The global minimum geodesic is the line $PQ$. There are infinitely many geodesics / locally shortest lines of course. –  Narasimham 2 days ago

Construct $\angle KOL$ "clockwise" of $\angle LOM$ and likewise $\angle JOK$ "clockwise" of $\angle KOL$ so that $m\angle JOK = m\angle KOL = m\angle LOM$. Then add images $P^*$ and $Q^*$ of $P$ and $Q$ so that $P^*$ and $Q^*$ are arranged the same way in $\angle JOK$ as $P$ and $Q$ are in $\angle LOM$.
The two path lengths in question are equal to $PQ^*$ and $P^*Q$. Use the law of cosines on $\triangle POQ^*$ and $\triangle P^*OQ$, along with the fact that cosine is monotonically decreasing on $(0, \pi)$, and the proposition should follow.