Let ABC be a triangle, C1 lie on BC. Circle (N) tangent AC1 at H, tangent tangent BC at M, tangent circumcircle ABC at T. Circle (N1) tangent AC at H1, tangent tangent BC at M1, tangent circumcircle (ABC1) at T1.
1-Prove that M1 ≡ M
2-Since http://www.cut-the-knot.org/Curriculum/Geometry/GeoGebra/OlegGolberg.shtml
We have AHIT cyclic and AE'H1T1 cyclic but problem is: (AHIT ) and (AE'H1T1 ) one circle. We away know I and E' => This problem show that away construct circle (N) and (N1)