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 4 of 9: Power of L gives a new cyclic quadrilateral

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LY\cdot LB=LP\cdot LM=LX\cdot LC\ \Longrightarrow\ B,Y,X,C\ \text{concyclic}

Detailed analysis

Since $L$ lies on line $AM$ , which also contains $P$ and $M$ , the power of $L$ with respect to $\odot(BPM)$ along the line through $B,Y$ (and $L$ ) and along the line through $P,M$ gives $LY\cdot LB=LP\cdot LM$ ; the power of $L$ with respect to $\odot(CPM)$ along line $C,X$ gives $LX\cdot LC=LP\cdot LM$ as well. Hence $LY\cdot LB=LX\cdot LC$ , so by the converse of power of a point, $B,Y,X,C$ are concyclic.