Circumcenter of a Triangle

Circumcenter of a Triangle

How to find the circumcenter of a triangle: 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 are the same.

Property 1:

The distances
between the circumcenter and each vertex
are the same.

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

Property 2:

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.

It says point O is the circumcenter.

So OM is the perpendicular bisector of BC.

So BM = MC.

See △OBM.

It's a right triangle.
And starting from the shortest side,
the sides are (3, BM, 5).

So △OBM is a (3, 4, 5) right triangle.

Pythagorean triples

So BM = 4.

BM = 4

And BM = MC.
So MC is also 4.

So BC = 4 + 4
= 8.