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 6 of 7: Circumcenter

O

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R,S\text{ projections of }O

Detailed analysis

Since $O$ is the circumcenter of $M^\prime BC$ and $N^\prime BC$ , the projections $R,S$ of $O$ to $HM,HN$ lie on the corresponding perpendicular bisectors. Similarity gives $HP/HR=HQ/HS$ .