Circumcenter of a Triangle

Circumcenter of a Triangle

How to solve the circumcenter of a triangle problems: definition, properties, example, and its solution.

Definition

The circumcenter of a triangle is the center of the circle that circumscribes the triangle.

The circumcenter of a triangle
is the center of the circle
that circumscribes the triangle.

Properties

The distances between the circumcenter and each vertex of a triangle are the same.

So the distances between the circumcenter
and each vertex of a triangle
are the same.

Three perpendicular bisectors of the triangle's sides meet at the circumcenter.

Three perpendicular bisectors of the triangle's sides
meet at the circumcenter.

Example

Point O is the circumcenter of triangle ABC. Find BC. OB = 5, OM = 3.

Point O is the circumcenter.
And OM (blue) is perpendicular to BC.

So OM is the perpendicular bisector of BC.
This means BM = MC.

OBM is a (3, 4, 5) right triangle.
So BM = 4.
Pythagorean triples

Then BM = MC = 4.

BM = MC = 4.

So BC = 4 + 4 = 8.