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 3 of 9: Reflect N over M to land back on AM

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L=BY\cap CX\ \Longrightarrow\ BNCL\ \text{is a parallelogram}\ \Longrightarrow\ L=2M-N\in AM

Detailed analysis

Let $L=BY\cap CX$ . Since $CL$ lies along $CX\parallel BD$ , $CL\parallel BN$ ; since $LB$ lies along $BY\parallel CE$ , $LB\parallel NC$ . So $BNCL$ has both pairs of opposite sides parallel, making it a parallelogram, and its diagonals $BC,NL$ bisect each other. As $M$ is the midpoint of $BC$ , $M$ is also the midpoint of $NL$ , so $L$ is the reflection of $N$ over $M$ . Since $N\in AM$ by step 1, its reflection $L$ over the point $M\in AM$ also lies on line $AM$ .