Mathematics

The key insight is that coaxial circles share a common radical axis, and the power of any point with respect to a circle determines the tangent length from that point. Since tangent length equals $\sqrt{\text{power}}$, you need to show that the ratio of powers from a point on one circle to two other circles in the system remains constant. This hinges on the radical axis property: power differences between circles are the same everywhere, so their ratio stabilizes as you move along any circle in the family.

The key insight is that coaxial circles share a common radical axis, and the power of any point with respect to a circle determines the tangent length from that point. Since tangent length equals $\sqrt{\text{power}}$, you need to show that the ratio of powers from a point on one circle to two other circles in the system remains constant. This hinges on the radical axis property: power differences between circles are the same everywhere, so their ratio stabilizes as you move along any circle in the family.