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Five points lie on a circle

Source: Iberoamerican 2016 P3

September 28, 2016

geometrycircumcircleIberoamericanIberoamerican 2016

Problem Statement

Let

ABC

be an acute triangle and

\Gamma

its circumcircle. The lines tangent to

\Gamma

through

B

and

C

meet at

P

. Let

M

be a point on the arc

AC

that does not contain

AM

such that

M \neq A

and

M \neq C

, and

K

be the point where the lines

BC

and

AM

meet. Let

R

be the point symmetrical to

P

with respect to the line

AM

and

Q

the point of intersection of lines

RA

and

PM

. Let

J

be the midpoint of

BC

and

L

be the intersection point of the line

PJ

and the line through

A

parallel to

PR

. Prove that

L, J, A, Q,

and

K

all lie on a circle.

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