Geometry • Topic 20

Radical Center

Just as the radical axis is the locus of equal power for two circles, the radical center is the point of equal power for three circles.

Definition

Definition (Radical Center)

For three circles with non-collinear centers, the radical center is the unique point

P

such that:

P_{ω_{1}} (P) = P_{ω_{2}} (P) = P_{ω_{3}} (P)

Theorem (Concurrency)

The radical axes of three circles (taken in pairs) are concurrent at the radical center.

Concurrency: Proving three lines intersect.
Orthocenter: The orthocenter of a triangle is the radical center of the three circles having the sides as diameters.
Excenters: The incenter is the radical center of the three excircles? (Check this).

Intersect any two radical axes. The third must pass through the same point.

Exercise (Problem 1)

Prove that the orthocenter of a triangle is the radical center of the three circles drawn with the altitudes as diameters.

Exercise (Problem 2)

Given three circles that are mutually tangent externally, prove that their common tangents at the points of contact are concurrent.

Exercise (Problem 3)

Find the coordinates of the radical center for circles:

x^{2} + y^{2} = 1

(x - 2)^{2} + y^{2} = 4

(x - 1)^{2} + (y - 2)^{2} = 9