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Nice collinearity!

Source: Romania TST 2014 Day 1 Problem 1

January 21, 2015

geometrypower of a pointradical axis

Problem Statement

Let

B

be a triangle, let

{A}'

,

{B}'

,

{C}'

be the orthogonal projections of the vertices

A

,

B

,

C

on the lines

BC

,

CA

and

AB

, respectively, and let

X

be a point on the line

A{A}'

.Let

\gamma_{B}

be the circle through

B

and

X

, centred on the line

BC

, and let

\gamma_{C}

be the circle through

C

and

X

, centred on the line

BC

.The circle

\gamma_{B}

meets the lines

AB

and

B{B}'

again at

M

and

{M}'

, respectively, and the circle

\gamma_{C}

meets the lines

AC

and

C{C}'

again at

N

and

{N}'

, respectively.Show that the points

M

,

{M}'

,

N

and

{N}'

are collinear.

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