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Nice geometry with many circles

Source: Greek BMO TST Problem 3

April 5, 2015

geometrycircumcircle

Problem Statement

Let

ABC

be an acute triangle with

\displaystyle{AB<AC<BC}

inscribed in circle

\displaystyle{c(O,R)}

.The excircle

\displaystyle{(c_A)}

has center

\displaystyle{I}

and touches the sides

\displaystyle{BC,AC,AB}

of the triangle

ABC

at

\displaystyle{D,E,Z}

respectively.

\displaystyle{AI}

cuts

\displaystyle{(c)}

at point

M

and the circumcircle

\displaystyle{(c_1)}

of triangle

\displaystyle{AZE}

cuts

\displaystyle{(c)}

at

K

.The circumcircle

\displaystyle{(c_2)}

of the triangle

\displaystyle{OKM}

cuts

\displaystyle{(c_1)}

at point

N

.Prove that the point of intersection of the lines

AN,KI

lies on

\displaystyle{(c)}

.

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