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 7 of 9: QRXY is cyclic

Previous step Next step

\angle PRQ=\angle PDE=\angle PXY\ \Longrightarrow\ Q,R,Y,X\ \text{concyclic}

Detailed analysis

By step 6, $D,P,Q$ and $E,P,R$ are collinear, so the directed angle $\angle PRQ=\angle PDE$ . By step 5, $XY\parallel DE$ , hence $\angle PDE=\angle PXY$ . Thus $\angle PRQ=\angle PXY$ , the equality of angles subtended by chord $QY$ at $R$ and $X$ ; therefore $Q,R,Y,X$ are cyclic.