MathDB

Consider an acute-angled triangle

\triangle ABC

(

AC>AB

) with its orthocenter

H

and circumcircle

\Gamma

.Points

M

P

are midpoints of

BC

and

AH

respectively.The line

\overline{AM}

meets

\Gamma

again at

X

and point

N

lies on the line

\overline{BC}

so that

\overline{NX}

is tangent to

\Gamma

. Points

J

and

K

lie on the circle with diameter

MP

such that

\angle AJP=\angle HNM

(

B

and

J

lie one the same side of

\overline{AH}

) and circle

\omega_1

, passing through

K,H

, and

J

, and circle

\omega_2

passing through

K,M

, and

N

, are externally tangent to each other. Prove that the common external tangents of

\omega_1

and

\omega_2

meet on the line

\overline{NH}

. Proposed by Alireza Dadgarnia

Problem Statement