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 7 of 7: Finish

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O,H,O_0\text{ collinear}

Detailed analysis

The line $HO_0$ meets the perpendiculars through $R,S$ at points whose distances from $H$ are equal by the displayed ratio; their intersection is $O$ . Hence $O_0$ lies on $OH$ .