Radical Centre

The point equidistant (in tangent length) from three circles. Found by solving any two of the three radical axis equations simultaneously.

Given three circles $S_1 = 0$, $S_2 = 0$, $S_3 = 0$, define three radical axes:

• $R_{12}$: radical axis of $S_1$ and $S_2$ → equation $S_1 - S_2 = 0$
• $R_{23}$: radical axis of $S_2$ and $S_3$ → equation $S_2 - S_3 = 0$
• $R_{13}$: radical axis of $S_1$ and $S_3$ → equation $S_1 - S_3 = 0$

Concurrence: Note that $(S_1 - S_2) + (S_2 - S_3) = S_1 - S_3$. So $R_{13}$ is a linear combination of $R_{12}$ and $R_{23}$. Any point on both $R_{12}$ and $R_{23}$ automatically satisfies $R_{13}$. Therefore the three radical axes meet at a single point — the radical centre.

At the radical centre $P$: $S_1(P) = S_2(P) = S_3(P)$, meaning the tangent lengths from $P$ to all three circles are equal.

Method: To find the radical centre, solve any two of the three equations $S_1 - S_2 = 0$ and $S_2 - S_3 = 0$ simultaneously. (Do not solve $S_1 = 0$ — that gives points on the circle, not the radical centre.)

Application: The radical centre is the centre of the unique circle that cuts all three given circles orthogonally.