I hope you could tell me if my reasoning is correct.
We are given two points $A$ and $B$ inside a rectangle $PQRS$. We create a path $AXYB$ such that $X$ and $Y$ lie on different sides of this rectangle. What is the minimal length of such path? How many shortest paths are there?
Given a line $\mathcal{l}$ and two points $A$ and $B$ on the same side of $\mathcal{l}$ we can construct the shortest path connecting them and touching $\mathcal{l}$ by reflecting for example $A$ in $\mathcal{l}$ (let $A'$ be the image of this reflection), connecting $A'$ and $B$, let $C$ be the point of intersection of $A'B$ with $\mathcal{l}$, and then $ACB$ is the shortest path.
Could we do something similar here?
I mean, could we find a point $X$ on one side of the rectangle connecting $A$ and $B$, and then $Y$ on another side connecting $X$ and $B$?
Does that make sense?
I really want to understand how this is done, because in the next exercise I'm asked to find the shortest path with two points inside a rectangle and three on its sides.
Best Answer
You can cover the plane in rectangles if you want, in a checkerboard pattern - alternating copies of the original and reflections of the original. You can use this to find the shortest path by locating the two points in each - reflecting in one side at a time.
Sometimes this is used in theoretical snooker, billiards or pool with tables which have perfect cushions and balls with no spin.