# How to find a minimal path without using a differential equation?

When browsing my son's math book, I stumbled upon a problem which I do not know how to approach at his level. This is a book for 11 years old, which is the 6th year of school ("6ème" in France).

The problem is the following

C and D are villages which are at, respectively 9 and 3 km from a railway (the EXF line).

They will build a station (denoted by X) on the railway in such a place where the distance to the villages CX + XD is the smallest.

Where should X be placed?

I took the easy way by creating a function which depends on EX and found the root of its derivative - which gave me the EX minimizing the distance.

Now the harder part is to find the solution appropriate for an 11 years old. Out of the possibly useful things they do not know yet are Pythagorean and Thales' theorems.

Let $D'$ be the reflection of $D$ across the railway line.
Then the distance from $X$ to $D'$ is the same as that from $X$ to $D$, and the shortest path from $C$ to $D'$ is a straight line. Place the station where this line intersects the railway line.
• From the measurements given you can determine the distance $XF$ in proportion to $EF$ ($\frac{XF}{EF}=\frac{1}{4}$) without resorting to the Pythagoras theorem or interpreting a map where $x$ marks the spot. Jun 11, 2018 at 20:45