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A Bunch of Tangencies

Source: OMM 2008 2

July 19, 2014

trigonometrygeometryperimeterinradiussimilar trianglesgeometry unsolved

Problem Statement

Consider a circle

\Gamma

, a point

A

on its exterior, and the points of tangency

B

and

C

from

A

to

\Gamma

. Let

P

be a point on the segment

AB

, distinct from

A

and

B

, and let

Q

be the point on

AC

such that

PQ

is tangent to

\Gamma

. Points

R

and

S

are on lines

AB

and

AC

, respectively, such that

PQ\parallel RS

and

RS

is tangent to

\Gamma

as well. Prove that

[APQ]\cdot[ARS]

does not depend on the placement of point

P

.

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