MathDB
ILL 92 nine-point circle

Source:

September 2, 2010
geometryTrianglecirclesIMO ShortlistIMO Longlist

Problem Statement

Let NN be a point inside the triangle ABCABC. Through the midpoints of the segments AN,BNAN, BN, and CNCN the lines parallel to the opposite sides of △ABC\triangle ABC are constructed. Let AN,BNAN, BN, and CNCN be the intersection points of these lines. If NN is the orthocenter of the triangle ABCABC, prove that the nine-point circles of △ABC\triangle ABC and △ANBNCN\triangle A_NB_NC_N coincide.
[hide="Remark."]Remark. The statement of the original problem was that the nine-point circles of the triangles ANBNCNA_NB_NC_N and AMBMCMA_MB_MC_M coincide, where NN and MM are the orthocenter and the centroid of ABCABC. This statement is false.
Back to ProblemsView on AoPS