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23

Part of 2014 Sharygin Geometry Olympiad

Problems(1)

A,B,C,D, triharmonic quadruple of points i.e. AB x CD = AC x BD= AD x BC

Source: Sharygin Geometry Olympiad 2014 Correspondence Round P23

8/1/2018
Let A,B,CA, B, C and DD be a triharmonic quadruple of points, i.e AB⋅CD=AC⋅BD=AD⋅BC.AB\cdot CD = AC \cdot BD = AD \cdot BC. Let A1A_1 be a point distinct from AA such that the quadruple A1,B,CA_1, B, C and DD is triharmonic. Points B1,C1B_1, C_1 and D1D_1 are defined similarly. Prove that a) A,B,C1,D1A, B, C_1, D_1 are concyclic; b) the quadruple A1,B1,C1,D1A_1, B_1, C_1, D_1 is triharmonic.
geometryharmonic divisionConcyclic