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 2 of 9: Chase angles to two parallel pairs

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\angle MCE=\angle MPE=\angle MPY=\angle MBY\ \Longrightarrow\ BY\parallel CE;\qquad \text{similarly}\quad CX\parallel BD

Detailed analysis

Since $C,M,P,E$ are concyclic, $\angle MCE=\angle MPE$ ; since $P,E,Y$ are collinear, $\angle MPE=\angle MPY$ ; since $B,M,P,Y$ are concyclic, $\angle MPY=\angle MBY$ . The equal directed angles $\angle MCE=\angle MBY$ that lines $CE$ and $BY$ make with line $BC$ (through $C$ and $B$ ) show $BY\parallel CE$ . The symmetric argument using $B,M,P,D$ and $D,P,X$ and $C,M,P,X$ concyclic/collinear gives $CX\parallel BD$ .