Mathematics

The key insight is recognizing that a circle touching two tangent lines and the original circle externally must have its center on the angle bisector from $P$, and that the director circle determines where $P$ can be. Since $P$ lies on the director circle, the angle $\angle APB$ is always $90°$, making the geometry symmetric. You'll need to use the relationship between the distances from the center of the small circle to the tangent lines and to the original circle's center to find its radius.

The key insight is recognizing that a circle touching two tangent lines and the original circle externally must have its center on the angle bisector from $P$, and that the director circle determines where $P$ can be. Since $P$ lies on the director circle, the angle $\angle APB$ is always $90°$, making the geometry symmetric. You'll need to use the relationship between the distances from the center of the small circle to the tangent lines and to the original circle's center to find its radius.