Problem 1

Let

H

be the orthocenter of triangle

ABC

. Let

M

and

N

be the midpoints of sides

AB

and

AC

, respectively. Assume that

H

lies inside quadrilateral

BMNC

and that the circumcircles of triangles

BMH

and

CNH

are tangent to each other. The line through

H

parallel to

BC

intersects the circumcircles of triangles

BMH

and

CNH

at points

K

and

L

, respectively. Let

F

be the intersection point of

MK

and

NL

, and let

J

be the incenter of triangle

MHN

. Prove that

FJ = FA

Step 2 of 7: Locate F inside triangle AMN

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F \in \operatorname{int}(\triangle AMN)

Detailed analysis

Because part of the circumcircle of $BMH$ lies outside triangle $ABC$ , the points $K$ and $N$ lie on opposite sides of line $AM$ ; likewise $L$ and $M$ lie on opposite sides of line $AN$ . Also $K$ and $L$ lie on the same side of $MN$ as $H$ , opposite to $A$ . These position facts place the intersection point $F$ of $MK$ and $NL$ inside triangle $AMN$ .