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Ah, don't bother me with this much point (show AE prp KL)

Source: Middle European Mathematical Olympiad 2011 - Individuals I-3

September 6, 2011

geometrycircumcirclegeometric transformationreflection

Problem Statement

In a plane the circles

\mathcal K_1

and

\mathcal K_2

with centers

I_1

and

I_2

, respectively, intersect in two points

A

and

B

. Assume that

\angle I_1AI_2

is obtuse. The tangent to

\mathcal K_1

in

A

intersects

\mathcal K_2

again in

C

and the tangent to

\mathcal K_2

in

A

intersects

\mathcal K_1

again in

D

. Let

\mathcal K_3

be the circumcircle of the triangle

BCD

. Let

E

be the midpoint of that arc

CD

of

\mathcal K_3

that contains

B

. The lines

AC

and

AD

intersect

\mathcal K_3

again in

K

and

L

, respectively. Prove that the line

AE

is perpendicular to

KL

.

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