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Unique point

Source: Greek national M.O. 2003, Final Round,problem 3

November 15, 2011

geometryratiogeometry unsolved

Problem Statement

Given are a circle

\mathcal{C}

with center

K

and radius

r,

point

A

on the circle and point

R

in its exterior. Consider a variable line

e

through

R

that intersects the circle at two points

B

and

C.

Let

H

be the orthocenter of triangle

ABC.

Show that there is a unique point

T

in the plane of circle

\mathcal{C}

such that the sum

HA^2 + HT^2

remains constant (as

e

varies.)

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