There isn't a single point that is equidistant from the towns. Explanation: See solution.
Practice makes perfect
If there is a point where the school can be placed so that it's equidistant from all of the towns, then we should be able to draw a circle that passes through all of the towns. To investigate this, we draw a triangle between three of the towns and find it's circumcenter.
To find the circumcenter, we have to graph at least two the perpendicular bisectors. We will start by constructing two circles, one around Cedar Falls and one around Shady Hills. Make sure the compass setting is the same for both circles and that the radius is greater than half the length of the side between the towns.
The perpendicular bisector is the segment where the two circles intersect.
Let's repeat the procedure for one more side.
Finally, we will plot the circumcenter and draw a circle using the circumcenter as our midpoint, and the distance between the circumcenter and one of the three towns as our radius.
As we can see, the locations that is equidistant from Cedar falls, Shady hills and Lake City is not equidistant from Willow Valley. Therefore, there is no location one can build the school to make it equidistant from all towns.
