The Radical Center of Three Circles

Radical Center

We know that every point on the radical axis has equal power with some two given circles. The next question that we can ask is what happens if we take three circles?

So, such point is the intersection of two of the axis if it exists, and then, the third radical axis must pass through this point!!

Voila!! You just proved that if we have three circles, then their pairwise radical axis pass through a point which has equal power with respect to both the circles, Or they don't intersect at all. That is they are parallel. Then again, parallel can be considered as intersecting at infinity.

Theorem Given any three circles such that no two are concentric, their pairwise radical axis are either concurrent or parallel.

You proved this theorem just a few moments ago!! Now we define this point of concurrency as Radical Center of three circles. In the figure below, the point $C$ denotes the radical center of the three circles in the figure.

The Radical Center is one of the most important and crucial properties of Radical Axis. It gives us a great tool to prove some difficult concurrency theorems in very elegant way.

If all the three pairwise radical axis of all the three circles are parallel, then their centers will be collinear. In a very special case where they actually have the same radical axis, we call these three circles Coaxial which will be dealt with in next topic.

Practice Problems coming up soon :D