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A crowded geometry

Source: Iran TST1 Day1 P2

February 22, 2020

geometryIranian TSTcircumcircle

Problem Statement

Let

O

be the circumcenter of the triangle

ABC

. Points

D,E

are on sides

AC,AB

and points

P,Q,R,S

are given in plane such that

P,C

and

R,C

are on different sides of

AB

and pints

Q,B

and

S,B

are on different sides of

AC

such that

R,S

lie on circumcircle of

DAP,EAQ

and

\triangle BCE \sim \triangle ADQ , \triangle CBD \sim \triangle AEP

(In that order),

\angle ARE=\angle ASD=\angle BAC

, If

RS\| PQ

prove that

RE ,DS

are concurrent on

AO

.
Proposed by Alireza Dadgarnia

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