Triangle: Circumcenter

How to use the property of the circumcenter of a triangle: definition, property, 1 example, and its solution.

Definition

Definition

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

Property

Property

Three perpendicular bisectors of the sides
meet at the circumcenter.

Example

Example

Solution

It says
point O is the circumcenter.

So, to remind yourself that
point O is the circumcenter,
lightly draw the circumscribed circle.

OM is perpendicular to BC.

Then OM is the perpendicular bisector of BC.

So BM = MC.

See △OBM.
It's a right triangle.

The sides are (3, BM, 5).
So △OBM is a (3, 4, 5) right triangle.

Pythagorean Triple

So BM = 4.

BM = MC
BM = 4

So MC = 4.

BM = 4
MC = 4

So BC = 4 + 4.

4 + 4 = 8

So BC = 8.