Geometry • Topic 9

Circumcenter

The circumcenter ( $O$ ) is the center of the circle that passes through all three vertices of a triangle.

Definition

Definition (Circumcenter)

The circumcenter is the intersection of the perpendicular bisectors of the three sides. It is the unique point equidistant from all three vertices (

O A = OB = OC = R

Location

Acute Triangle: Inside the triangle.
Right Triangle: Midpoint of the hypotenuse.
Obtuse Triangle: Outside the triangle.

Key Properties

1. Extended Law of Sines

The circumradius

R

links side lengths and angles:

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C} = 2 R

2. Coordinate Geometry

If the origin is the circumcenter, then

∣ A ∣ = ∣ B ∣ = ∣ C ∣ = R

3. Angle Relations

The angle at the center is double the angle at the circumference:

$∠ BOC = 2∠ A$ (if $A$ is acute).
If $A$ is obtuse, the reflex angle is $2 A$ .

Practice Problems

Exercise (Problem 1)

△ A BC

A B = A C = 5

and

BC = 6

. Find the circumradius

R

Exercise (Problem 2)

Let

O

be the circumcenter of

△ A BC

. If

∠ O A B = 2 5^{\circ}

, find

∠ C

Exercise (Problem 3)

Prove that the reflection of the orthocenter across any side lies on the circumcircle.