The Perpendicular Bisectors and the circumcenter of a Triangle

The perpendicular bisectors of the sides of a triangle meet in a common intersection point. This point is called the circumcenter. The circumcenter is the center of the circumscribed circle.

Below we see that the perpendicular bisectors always intersect in a point. If you select one of the vertices of the triangle, you can alter the triangle and show that the perpendicular bisectors of this new triangle still intersect in 1 point. Furthermore, we can show that the circumcenter is always the center of the circumscribed circle.

When does the circumcenter lie inside the triangle?
When does the circumcenter lie outside the triangle?
When does the circumcenter lie on one of the sides of the triangle?

{1} Point(93,210)[label('A')]; {2} Point(275,113)[label('B')]; {3} Point(139,71)[label('C')]; {4} Segment(2,1)[thick,black]; {5} Segment(3,2)[thick,black]; {6} Segment(1,3)[thick,black]; {7} Midpoint(6); {8} Midpoint(4); {9} Midpoint(5); {10} Perpendicular(6,7)[blue]; {11} Perpendicular(4,8)[blue]; {12} Perpendicular(5,9)[blue]; {13} Intersect(12,10)[label('CENTROID')]; {14} Intersect(5,11); {15} Circle(13,2)[green]; "> Sorry, this page requires a Java-compatible web browser.

Written by Dr. Anneke Bart, Saint Louis University. For comments email me at: barta@slu.edu