MathDB

Vietnam TST #3

Source: Vietnam TST 2022 P3

April 26, 2022

geometryparallelogramcircumcircle

Problem Statement

Let

ABCD

be a parallelogram,

AC

intersects

BD

at

I

. Consider point

G

inside

\triangle ABC

that satisfy

\angle IAG=\angle IBG\neq 45^{\circ}-\frac{\angle AIB}{4}

. Let

E,G

be projections of

C

on

AG

and

D

on

BG

. The

E-

median line of

\triangle BEF

and

F-

median line of

\triangle AEF

intersects at

H

.

a)

Prove that

AF,BE

and

IH

concurrent. Call the concurrent point

L

.

b)

Let

K

be the intersection of

CE

and

DF

. Let

J

circumcenter of

(LAB)

and

M,N

are respectively be circumcenters of

(EIJ)

and

(FIJ)

. Prove that

EM,FN

and the line go through circumcenters of

(GAB),(KCD)

are concurrent.

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