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Hard Geometry on circle intersections

Source: IMEO 2019, Problem 6

October 14, 2019

geometrygeometry proposedcircle intersections

Problem Statement

Let

ABC

be a scalene triangle with incenter

I

and circumcircle

\omega

. The internal and external bisectors of angle

\angle BAC

intersect

BC

at

D

and

E

, respectively. Let

M

be the point on segment

AC

such that

MC = MB

. The tangent to

\omega

at

B

meets

MD

at

S

. The circumcircles of triangles

ADE

and

BIC

intersect each other at

P

and

Q

. If

AS

meets

\omega

at a point

K

other than

A

, prove that

K

lies on

PQ

.
Proposed by Alexandru Lopotenco (Moldova)

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