Problem 4

Let

ABC

be an acute triangle with

AB>BC

and

AC>BC

. Let

O,H

be its circumcenter and orthocenter. The circumcircle of

AHC

meets line

AB

again at

M

, and the circumcircle of

AHB

meets line

AC

again at

N

. Prove that the circumcenter of triangle

MNH

lies on line

OH

Step 1 of 7: Auxiliary points

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M,M^\prime,C,H\text{ concyclic}

Detailed analysis

Let $M^\prime$ be the second intersection of $CH$ with the circumcircle of $ABC$ , and $N^\prime$ the second intersection of $BH$ with that circle. Cyclic angles show quadrilaterals $HBM^\prime M$ and $HCN^\prime N$ are rhombi.