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Circle + ratios

Source: Argentina IMO TST Problem 2

August 27, 2009

ratiogeometryalgebra proposedalgebra

Problem Statement

Triangle

ABC

is inscript in a circumference

Y

. A chord MN\equal{}1 of

N

intersects the sides

MN

and

AC

at

X

and

Y

respectively, with

U

,

A

,

MN

,

N

in that order in

MN

. Let

UV

be the diameter of

\Gamma

perpendicular to

MN

with

U

and

A

in the same semiplane respect to

MN

. Lines

AV

,

BU

and

CU

cut

MN

in the ratios

\frac{3}{2}

,

\frac{4}{5}

and

\frac{7}{6}

respectively (starting counting from

M

). Find

XY

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