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A difficult problem [tangent circles in right triangles]

Source: IMO ShortList 1998, geometry problem 8; Yugoslav TST 1999

October 17, 2004

geometrycircumcirclereflectioncomplex numbersIMO Shortlistgeometry solvedharmonic division

Problem Statement

Let

ABC

be a triangle such that

\angle A=90^{\circ }

and

\angle B<\angle C

. The tangent at

A

to the circumcircle

\omega

of triangle

ABC

meets the line

BC

at

D

. Let

E

be the reflection of

A

in the line

BC

, let

X

be the foot of the perpendicular from

A

to

BE

, and let

Y

be the midpoint of the segment

AX

. Let the line

BY

intersect the circle

\omega

again at

Z

. Prove that the line

BD

is tangent to the circumcircle of triangle

ADZ

. [hide="comment"] Edited by Orl.

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