A circle
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
bisects a circle
![S'](/media/m/f/4/5/f457ce3c1c610a827678f4d2c4464cda.png)
if it cuts
![S'](/media/m/f/4/5/f457ce3c1c610a827678f4d2c4464cda.png)
at opposite ends of a diameter.
![S_A](/media/m/7/f/c/7fcc8745e93c4d21f42b0f1b226f7ca8.png)
,
![S_B](/media/m/6/4/4/6447f0af6ea094ff011452733cfaf875.png)
,
![S_C](/media/m/9/3/4/93489b4c83f0e88f5d27ef25a4f90eb6.png)
are circles with distinct centers
![A, B, C](/media/m/5/2/5/5251ced8c37ecf5247e7f644e571612f.png)
(respectively).
Show that
![A, B, C](/media/m/5/2/5/5251ced8c37ecf5247e7f644e571612f.png)
are collinear iff there is no unique circle
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
which bisects each of
![S_A](/media/m/7/f/c/7fcc8745e93c4d21f42b0f1b226f7ca8.png)
,
![S_B](/media/m/6/4/4/6447f0af6ea094ff011452733cfaf875.png)
,
![S_C](/media/m/9/3/4/93489b4c83f0e88f5d27ef25a4f90eb6.png)
. Show that if there is more than one circle
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
which bisects each of
![S_A](/media/m/7/f/c/7fcc8745e93c4d21f42b0f1b226f7ca8.png)
,
![S_B](/media/m/6/4/4/6447f0af6ea094ff011452733cfaf875.png)
,
![S_C](/media/m/9/3/4/93489b4c83f0e88f5d27ef25a4f90eb6.png)
, then all such circles pass through two fixed points. Find these points.
Original Statement:
A circle
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
is said to cut a circle
![\Sigma](/media/m/c/a/b/cab3b5d26e6dabb66878a77b1791db45.png)
diametrically if and only if their common chord is a diameter of
Let
![S_A, S_B, S_C](/media/m/5/5/a/55a04fa4ecb918375c3954b1e90e98d7.png)
be three circles with distinct centres
![A,B,C](/media/m/6/0/1/6012c28979f41c54e9b40b9fc855aa34.png)
respectively. Prove that
![A,B,C](/media/m/6/0/1/6012c28979f41c54e9b40b9fc855aa34.png)
are collinear if and only if there is no unique circle
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
which cuts each of
![S_A, S_B, S_C](/media/m/5/5/a/55a04fa4ecb918375c3954b1e90e98d7.png)
diametrically. Prove further that if there exists more than one circle
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
which cuts each
![S_A, S_B, S_C](/media/m/5/5/a/55a04fa4ecb918375c3954b1e90e98d7.png)
diametrically, then all such circles
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
pass through two fixed points. Locate these points in relation to the circles
%V0
A circle $S$ bisects a circle $S'$ if it cuts $S'$ at opposite ends of a diameter. $S_A$, $S_B$,$S_C$ are circles with distinct centers $A, B, C$ (respectively).
Show that $A, B, C$ are collinear iff there is no unique circle $S$ which bisects each of $S_A$, $S_B$,$S_C$ . Show that if there is more than one circle $S$ which bisects each of $S_A$, $S_B$,$S_C$ , then all such circles pass through two fixed points. Find these points.
Original Statement:
A circle $S$ is said to cut a circle $\Sigma$ diametrically if and only if their common chord is a diameter of $\Sigma.$
Let $S_A, S_B, S_C$ be three circles with distinct centres $A,B,C$ respectively. Prove that $A,B,C$ are collinear if and only if there is no unique circle $S$ which cuts each of $S_A, S_B, S_C$ diametrically. Prove further that if there exists more than one circle $S$ which cuts each $S_A, S_B, S_C$ diametrically, then all such circles $S$ pass through two fixed points. Locate these points in relation to the circles $S_A, S_B, S_C.$