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prove 5 point lie on a circle

Source: Vietnam TST 2000

April 3, 2007

geometrycircumcircle

Problem Statement

Two circles

C_{1}

and

C_{2}

intersect at points

P

and

Q

. Their common tangent, closer to

P

than to

Q

, touches

C_{1}

at

A

and

C_{2}

at

B

. The tangents to

C_{1}

and

C_{2}

at

P

meet the other circle at points

E \not = P

and

F \not = P

, respectively. Let

H

and

K

be the points on the rays

AF

and

BE

respectively such that

AH = AP

and

BK = BP

. Prove that

A,H,Q,K,B

lie on a circle.

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