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 4 of 7: Similarity

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HBM^\prime M\sim HCN^\prime N

Detailed analysis

The rhombi are composed of right triangles with the same acute angle, hence the similarity carries $HM$ to $HN$ and the perpendicular bisector of $BM^\prime$ to that of $CN^\prime$ .