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All such circles pass through two fixed points

Source: IMO Shortlist 1993, United Kingdom 1

October 24, 2005
geometrypower of a pointperpendicular bisectorcirclesIMO Shortlist

Problem Statement

A circle SS bisects a circle S′S' if it cuts S′S' at opposite ends of a diameter. SAS_A, SBS_B,SCS_C are circles with distinct centers A,B,CA, B, C (respectively). Show that A,B,CA, B, C are collinear iff there is no unique circle SS which bisects each of SAS_A, SBS_B,SCS_C . Show that if there is more than one circle SS which bisects each of SAS_A, SBS_B,SCS_C , then all such circles pass through two fixed points. Find these points. Original Statement: A circle SS is said to cut a circle Σ\Sigma diametrically if and only if their common chord is a diameter of Σ.\Sigma. Let SA,SB,SCS_A, S_B, S_C be three circles with distinct centres A,B,CA,B,C respectively. Prove that A,B,CA,B,C are collinear if and only if there is no unique circle SS which cuts each of SA,SB,SCS_A, S_B, S_C diametrically. Prove further that if there exists more than one circle SS which cuts each SA,SB,SCS_A, S_B, S_C diametrically, then all such circles SS pass through two fixed points. Locate these points in relation to the circles SA,SB,SC.S_A, S_B, S_C.
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