MathDB

Let

A, B, C, A', B', C'

be six pairs of different points. Prove that the Circles

BCA'

CAB'

and

ABC'

have a common point, then the Circles

B'C'A, C'A'B

and

A'B'C

also share a common point.
Note: For three pairs of different points

X, Y

and

Z

we define the Circle

XYZ

as the circumcircle of the triangle

XYZ

, or - in the case when the points

X, Y

and

Z

lie on a straight line - this straight line.

Problem Statement