Radical axis

The radical axis (or power line) of two circles is the locus of points at which tangents drawn to both circles have the same length. The radical axis is always a straight line and always perpendicular to the line connecting the centers of the circles, albeit closer to the circumference of the larger circle. If the circles intersect, the radical axis is the line passing through the intersection points; similarly, if the circles are tangent, the radical axis is simply the common tangent. The radical axis (or power line) of two circles is the locus of points at which tangents drawn to both circles have the same length. The radical axis is always a straight line and always perpendicular to the line connecting the centers of the circles, albeit closer to the circumference of the larger circle. If the circles intersect, the radical axis is the line passing through the intersection points; similarly, if the circles are tangent, the radical axis is simply the common tangent. For any point P on the radical axis, there is a unique circle centered on P that intersects both circles at right angles (orthogonally); conversely, the center of any circle that cuts both circles orthogonally must lie on the radical axis. In technical language, each point P on the radical axis has the same power with respect to both circles where r1 and r2 are the radii of the two circles, d1 and d2 are distances from P to the centers of the two circles, and R is the radius of the unique orthogonal circle centered on P. In general, two disjoint, non-concentric circles can be aligned with the circles of bipolar coordinates; in that case, the radical axis is simply the y-axis; every circle on that axis that passes through the two foci intersect the two circles orthogonally. Thus, two radii of such a circle are tangent to both circles, satisfying the definition of the radical axis. The collection of all circles with the same radical axis and with centers on the same line is known as a pencil of coaxal circles. Consider three circles A, B and C, no two of which are concentric. The radical axis theorem states that the three radical axes (for each pair of circles) intersect in one point called the radical center, or are parallel. In technical language, the three radical axes are concurrent (share a common point); if they are parallel, they concur at a point of infinity. A simple proof is as follows. The radical axis of circles A and B is defined as the line along which the tangents to those circles are equal in length a=b. Similarly, the tangents to circles B and C must be equal in length on their radical axis. By the transitivity of equality, all three tangents are equal a=b=c at the intersection point r of those two radical axes. Hence, the radical axis for circles A and C must pass through the same point r, since a=c there. This common intersection point r is the radical center.