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 9 of 9: The fixed point T lies on circle AXY

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T=AS\cap\Omega\ (T\ne A),\ \text{fixed};\qquad SX\cdot SY=SB\cdot SC=ST\cdot SA\ \Longrightarrow\ A,X,Y,T\ \text{concyclic}

Detailed analysis

Let $T$ be the second intersection of line $AS$ with $\Omega$ ; since $S$ is fixed (step 8) and $A,\Omega$ are fixed, $T$ is fixed. The power of $S$ with respect to $\Omega$ gives $ST\cdot SA=SB\cdot SC$ ; the power of $S$ with respect to $\odot(BCXY)$ gives $SB\cdot SC=SX\cdot SY$ . Hence $SX\cdot SY=ST\cdot SA$ , so by the converse of power of a point, $A,T,X,Y$ are concyclic. Thus the circumcircle of $AXY$ always passes through the fixed point $T\ne A$ .