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 8 of 9: A second radical center is fixed

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S=\text{radical center of }\Omega,\ \odot(BCXY),\ \odot(QRXY)\ \Longrightarrow\ S=BC\cap QR\cap XY,\ \text{fixed}

Detailed analysis

Let $S$ be the radical center of $\Omega$ , $\odot(BCXY)$ (step 4), and $\odot(QRXY)$ (step 7). The pairwise radical axes are $BC$ (common points of $\Omega,\odot(BCXY)$ ), $QR$ (common points of $\Omega,\odot(QRXY)$ , as $Q,R\in\Omega$ ), and $XY$ (common points of $\odot(BCXY),\odot(QRXY)$ ), so $S=BC\cap QR\cap XY$ . Since $Q,R$ are fixed (step 6) and $B,C$ are fixed vertices, the point $S=BC\cap QR$ does not depend on $P$ .