Radical Center of Three 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.
At the radical center, there is a unique circle that is orthogonal to all three circles. This follows because each radical axis is the locus of centers of circles that cut each pair of given circles orthogonally.
Read more about this topic: Radical Axis
Famous quotes containing the words radical, center and/or circles:
“Any critic is entitled to wrong judgments, of course. But certain lapses of judgment indicate the radical failure of an entire sensibility.”
—Susan Sontag (b. 1933)
“New York is what Paris was in the twenties ... the center of the art world. And we want to be in the center. It’s the greatest place on earth.... I’ve got a lot of friends here and I even brought my own cash.”
—John Lennon (1940–1980)
“Before the birth of the New Woman the country was not an intellectual desert, as she is apt to suppose. There were teachers of the highest grade, and libraries, and countless circles in our towns and villages of scholarly, leisurely folk, who loved books, and music, and Nature, and lived much apart with them. The mad craze for money, which clutches at our souls to-day as la grippe does at our bodies, was hardly known then.”
—Rebecca Harding Davis (1831–1910)