Question

Three highways connect the centers of three towns and form a triangle. A cell phone company wants to place a new cell tower so that it is the same distance from the centers of the three towns. How can the company find where to place the tower? Explain.

Answer 1

The point that is the same distance from the three town centers (the triangle's vertices) is the triangle's circumcenter — the center of the circle passing through all three vertices.

How to find it (compass-and-straightedge / construction):
1. Label the towns A, B, C.
2. Construct the perpendicular bisector of segment AB:
- With center A and B and radius greater than half AB, draw two arcs that intersect in two points.
- Draw the line through these two arc-intersection points; that line is the perpendicular bisector of AB.
3. Construct the perpendicular bisector of segment BC in the same way.
4. The intersection point O of those two perpendicular bisectors is the circumcenter. Place the tower at O.

Why this works:
- Every point on the perpendicular bisector of AB is equidistant from A and B.
- The intersection of the perpendicular bisectors of two sides is therefore equidistant from A, B, and C, so it is equidistant from all three town centers.

Note: For an acute triangle the circumcenter lies inside the triangle (tower inside), for a right triangle it is the midpoint of the hypotenuse, and for an obtuse triangle it lies outside the triangle.