Optimal Placement of Water Fountain in a City: Solving for Equidistant Points

[moonman239]

Let's say a city planner plans to place a fountain that is equidistant from where people will want to walk toward a water fountain (e.g, where a runner will likely stop along a trail and turn towards a water fountain). He identifies 7 such spots.

How would the builder go about finding the spot for a water fountain?

 

[HallsofIvy]

Set up a coordinate system giving (x,y) values to each of the points. The "centroid" will be the point whose x coordinate is the average of the 7 x values of the points and y coordinate will be the average of the 7 y values of the points.