Question

Short answer

It is not possible to find a spot for the recreation centre which is equidistant from three towns.

Step-by-step solution

Step 1. Given information:

Three points $X,Y,Z$ as collinear points.

Step 2. Concept used.

Centroid and median relation of a triangle.

Step 3. applying the concept.

Consider the segments.

Draw the perpendicular bisector of each segment as shown in the figure:

We know that perpendicular bisectors are parallel to each other.