MathDB

20

Part of 2006 Sharygin Geometry Olympiad

Problems(1)

concurrent circles passing through midpoints of triangle's sides (orthocenter)

Source: Sharygin 2006 X-XI CR 20

8/26/2019
Four points are given A,B,C,DA, B, C, D. Points A1,B1,C1,D1A_1, B_1, C_1,D_1 are orthocenters of the triangles BCD,CDA,DAB,ABCBCD, CDA, DAB, ABC and A2,B2,C2,D2A_2, B_2, C_2,D_2 are orthocenters of the triangles B1C1D1,C1D1A1,D1A1B1,A1B1C1B_1C_1D_1, C_1D_1A_1, D_1A_1B_1,A_1B_1C_1 etc. Prove that the circles passing through the midpoints of the sides of all the triangles intersect at one point.
orthocentergeometrycirclesmidpointconcurrencyconcurrent