Question 21
(GBR 1) A circle $S$ is said to cut a circle $\Sigma$ diametrally if their common chord is a diameter of $\Sigma$.
Let $S_A,S_B,S_C$ be three circles with distinct centers $A,B,C$ respectively. Prove that $A,B,C$ are collinear if and only if there is no unique circle $S$ that cuts each of $S_A,S_B,S_C$ diametrally. Prove further that if there exists more than one circle $S$ that cuts each of $S_A,S_B,S_C$ diametrally, then all such circles pass through two fixed points. Locate these points in relation to the circles $S_A,S_B,S_C$.
Let $S_A,S_B,S_C$ be three circles with distinct centers $A,B,C$ respectively. Prove that $A,B,C$ are collinear if and only if there is no unique circle $S$ that cuts each of $S_A,S_B,S_C$ diametrally. Prove further that if there exists more than one circle $S$ that cuts each of $S_A,S_B,S_C$ diametrally, then all such circles pass through two fixed points. Locate these points in relation to the circles $S_A,S_B,S_C$.