Geometry / Unit 6: Relationships in Triangles
6.3 Circumcenter and Incenter
Notes
Three bisectors of a triangle always meet at one point (they are concurrent).
- The perpendicular bisectors meet at the circumcenter. It is equidistant from the three vertices, so it is the center of the circle through them.
- The angle bisectors meet at the incenter. It is equidistant from the three sides, so it is the center of the circle inside touching all sides.
- The incenter is always inside. The circumcenter is inside an acute triangle, on the hypotenuse's midpoint for a right triangle, and outside an obtuse triangle.
Practice
Which point of a triangle is equidistant from all three vertices?
The circumcenter lies on all three perpendicular bisectors.
A right triangle has a hypotenuse of length 26. How far is the circumcenter from each vertex?
The circumcenter is the midpoint of the hypotenuse, so the distance is $26 \div 2 = 13$.
A city wants a fountain the same distance from three straight roads that form a triangle. Where should it go?
The incenter is equidistant from the three sides.
Ready for more? Take the Unit 6 test.