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Turkey NMO 1999, P-2,a relation on a circle

Source:

December 23, 2010

geometry proposedgeometry

Problem Statement

Problem-2: Given a circle with center

S

, the two tangent lines from a point

X

outside the circle touch the circle at points

PB

and

Q

. Line

SO

intersects the circle at

A

and

B

, with

B

closer to

S

. Let

X

be an interior point of minor arc

PB

, and let line

OS

intersect lines

QX

and

PX

at

C

and

D

, respectively. Prove that

\frac{1}{\left| AC \right|}+\frac{1}{\left| AD \right|}=\frac{2}{\left| AB \right|}

.

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