Problem 3

Let

ABC

be a scalene triangle with circumcircle

\Omega

. Let

M

be the midpoint of

BC

. A variable point

P

is selected on segment

AM

. The circumcircles of triangles

BPM

and

CPM

meet

\Omega

again at points

D

and

E

, respectively. The lines

DP

and

EP

meet the circumcircles of triangles

CPM

and

BPM

again at points

X

and

Y

, respectively. Prove that, as

P

varies, the circumcircle of triangle

AXY

passes through a fixed point

T

distinct from

A

Step 1 of 9: Locate the first radical center on line AM

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N=\text{radical center of }\Omega,\ \odot(BPM),\ \odot(CPM)\ \Longrightarrow\ BD,\ CE,\ PM\ \text{concur at}\ N,\quad N\in AM

Detailed analysis

Let $N$ be the radical center of $\Omega$ , $\odot(BPM)$ , and $\odot(CPM)$ . The pairwise radical axes are $BD$ (common points of $\Omega,\odot(BPM)$ ), $CE$ (common points of $\Omega,\odot(CPM)$ ), and $PM$ (common points of $\odot(BPM),\odot(CPM)$ ), so these three lines concur at $N$ . Since $P$ and $M$ both lie on segment $AM$ , line $PM$ is exactly line $AM$ , so $N\in AM$ .