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A geometry problem from VMO 2005

Source: Vietnam MO 2005,problem2

March 10, 2005

geometryparallelogramtrapezoidLaTeXgeometric transformationreflectioninequalities

Problem Statement

Let

(O)

be a fixed circle with the radius

R

. Let

A

and

B

be fixed points in

(O)

such that

A,B,O

are not collinear. Consider a variable point

C

lying on

(O)

(

C\neq A,B

). Construct two circles

(O_1),(O_2)

passing through

A,B

and tangent to

BC,AC

at

C

, respectively. The circle

(O_1)

intersects the circle

(O_2)

in

D

(

D\neq C

). Prove that: a)

CD\leq R

b) The line

CD

passes through a point independent of

C

(i.e. there exists a fixed point on the line

CD

when

C

lies on

(O)

).

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