Problem
GEO-B2-M08-P024 Assemble the Configuration Yourself
Four lines in general position are given. Four circles are constructed on the triangles formed by triples of these lines. Prove that if three of these circles have a common point \(M\), then the fourth circle also passes through \(M\).
Label the six points \(A,B,C,D,E,F\) and reduce the problem to Miquel's theorem.
Label the intersection points as \(A,B,C,D,E,F\). The four circles necessarily have the form \((ABE)\), \((ADF)\), \((BCF)\), \((CDE)\). If three of them pass through \(M\), then by choosing two circles with a common vertex, we get \(M\) as the Miquel point of the complete quadrilateral. By Miquel's theorem, \(M\) also lies on the fourth circle.
The final problem checks whether the student can recognise the complete quadrilateral independently.