Problem
GEO-B2-M05-P024 Choosing the Centre at a Common Point
Three circles pass through one point \(O\). Each pair meets again at points \(A\), \(B\), \(C\): \(\omega_1\cap\omega_2=\{O,A\}\), \(\omega_2\cap\omega_3=\{O,B\}\), \(\omega_3\cap\omega_1=\{O,C\}\). Perform an inversion with centre \(O\). Describe the image configuration.
Each circle through \(O\) becomes a line.
Circles \(\omega_1,\omega_2,\omega_3\) pass through the centre of inversion, so their images are three lines \(l_1,l_2,l_3\). Point \(A\) lies on \(\omega_1\) and \(\omega_2\), hence \(A'\) lies on \(l_1\) and \(l_2\). Similarly, \(B'=l_2\cap l_3\), and \(C'=l_3\cap l_1\). Thus the configuration of three circles through \(O\) becomes a triangle formed by three lines.
The final problem shows the strategy of choosing the inversion centre.