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China Mathematics Olympiads (National Round) 2008 Problem 1

Source:

November 28, 2010

geometrycircumcircleincentergeometric transformationreflectionperpendicular bisectorgeometry unsolved

Problem Statement

Suppose

\triangle ABC

is scalene.

O

is the circumcenter and

A'

is a point on the extension of segment

AO

such that

\angle BA'A = \angle CA'A

. Let point

A_1

and

A_2

be foot of perpendicular from

A'

onto

AB

and

AC

.

H_{A}

is the foot of perpendicular from

A

onto

BC

. Denote

R_{A}

to be the radius of circumcircle of

\triangle H_{A}A_1A_2

. Similiarly we can define

R_{B}

and

R_{C}

. Show that:

\frac{1}{R_{A}} + \frac{1}{R_{B}} + \frac{1}{R_{C}} = \frac{2}{R}

where R is the radius of circumcircle of

\triangle ABC

.

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