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collinear wanted, PQ= PR, 3 circles related

Source: Mediterranean Mathematical Olympiad 2020 P4 MMC

September 23, 2020

geometrycollinear

Problem Statement

Let

P,Q,R

be three points on a circle

k_1

with

|PQ|=|PR|

and

|PQ|>|QR|

. Let

k_2

be the circle with center in

P

that goes through

Q

and

R

. The circle with center

Q

through

Y\ne R

intersects

k_1

in another point

X\ne R

and intersects

k_2

in another point

Y\ne R

. The two points

X

and

R

lie on different sides of the line through

PQ

. Show that the three points

P

,

X

,

Y

lie on a common line.

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