Problem 2 Using a Circumcenter

A town planner wants to locate a new fire station equidistant from the elementary, middle, and high schools. Where should he locate the station?

Image Long Description

The three schools form the vertices of a triangle. The planner should locate the fire station at P, the point of concurrency of the perpendicular bisectors of Δ E M H . cap delta e m h .  This point is the circumcenter of Δ E M H cap delta e m h  and is equidistant from the three schools at E, M, and H.

✓ Got It?

2. In Problem 2, the town planner wants to place a bench equidistant from the three trees in the park. Where should he place the bench?

Take Note Theorem 5-7: Concurrency of Angle Bisectors Theorem

Theorem

The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle.

Diagram

Symbols

Angle bisectors A P ¯ , B P ¯ , eh p bar , comma . b p bar , comma  and C P ¯ c p bar  are concurrent at P.
P X = P Y = P Z p x equals . p y equals p z

You will prove Theorem 5-7 in Exercise 24.