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Vietnam TST 2017 problem 3

Source: Vietnam TST 2017

March 26, 2017

geometry

Problem Statement

Triangle

ABC

with incircle

(I)

touches the sides

AB, BC, AC

at

F, D, E

, res.

I_b, I_c

are

B

- and

C-

excenters of

ABC

.

P, Q

are midpoints of

I_bE, I_cF

.

(PAC)\cap AB=\{ A, R\}

,

(QAB)\cap AC=\{ A,S\}

. a. Prove that

PR, QS, AI

are concurrent. b.

DE, DF

cut

I_bI_c

at

K, J

, res.

EJ\cap FK=\{ M\}

.

PE, QF

cut

(PAC), (QAB)

at

X, Y

res. Prove that

BY, CX, AM

are concurrent.

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