MathLabs

Problem 4

Let ABCABC be an acute triangle with AB>BCAB>BC and AC>BCAC>BC. Let O,HO,H be its circumcenter and orthocenter. The circumcircle of AHCAHC meets line ABAB again at MM, and the circumcircle of AHBAHB meets line ACAC again at NN. Prove that the circumcenter of triangle MNHMNH lies on line OHOH.
Step 2 of 7: First rhombus
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HM=HB=BM′=M′HHM=HB=BM^\prime=M^\prime H
Detailed analysis

From the two cyclic quadrilaterals and the perpendicular relations through the orthocenter, the four sides of HBM′MHBM^\prime M are equal.