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 5 of 7: Circumcenter O0O_0
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P,Q midpoints of HM,HNP,Q\text{ midpoints of }HM,HN
Detailed analysis

If O0O_0 is the circumcenter of MNHMNH, its perpendicular projections to HM,HNHM,HN are their midpoints P,QP,Q.