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Proving collinearity

Source: Bulgarian IMO TST 2008, Day 2, Problem 2

July 8, 2013

geometrycircumcirclegeometric transformationhomothetytrigonometrypower of a pointradical axis

Problem Statement

In the triangle

ABC

,

AM

is median,

M \in BC

,

BB_{1}

and

CC_{1}

are altitudes,

BB_{1}

,

B_{1} \in AC

. The line through

A

which is perpendicular to

AM

cuts the lines

BB_{1}

and

CC_{1}

at points

E

and

F

, respectively. Let

k

be the circumcircle of

\triangle EFM

. Suppose also that

k_{1}

and

k_{2}

are circles touching both

EF

and the arc

EF

of

k

which does not contain

M

. If

P

and

Q

are the points at which

k_{1}

intersects

k_{2}

, prove that

P

,

Q

, and

M

are collinear.

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