Problem 1

Let

ABC

be an acute-angled triangle with

AB\ne AC

. The circle with diameter

BC

intersects the sides

AB

and

AC

M

and

N

respectively. Denote by

O

the midpoint of side

BC

. The bisectors of angles

\angle BAC

and

\angle MON

intersect at

R

. Prove that the circumcircles of triangles

BMR

and

CNR

have a common point lying on side

BC

Step 2 of 4: Two auxiliary circles

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\angle AMH=\angle ANH=90^\circ \implies A,M,H,N\text{ concyclic (circle on diameter }AH\text{)}

Detailed analysis

The lines $CM$ and $BN$ are perpendicular to $AB$ and $AC$ , respectively. Hence $angle AMH=angle ANH=90^circ$ , so $A,M,H,N$ lie on the circle with diameter $AH$ . This auxiliary circle will identify the point $R$ .