Proof: Theorem 4.5
Proof: Theorem 4.5
Theorem 4.5: If three circles are given with noncollinear centers and each pair of circles determines a radical axis, then three radical axes are concurrent.
Proof: Assume A, B, and C are circles with non-collinear centers. Let be the center of A, be the center of B, and be the center of C. Let B intersect both A and C. In order to find the radical axes of each pair of circles, we must first determine the points of intersection between all circles. Note that the points of intersection for circles A and B are points m and n. Similarly, the points of intersection between B and C are points o and p. By definition of radical axis, we know that line is the radical axis of circles A and B. Similarly, we know that is the radical axis of circles B and C. Construct point X which is the intersection of and . Since circles A and C do not intersect, start by constructing line t through and . Then construct line that is perpendicular to t and passes through point X. Thus, all radical axes pass through point X. Thus, , , and are concurrent.