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austrian collinear, incircle and midpoints related

Source: 49th Austrian Mathematical Olympiad National Competition (Final Round ) 28th April 2018 p2

May 25, 2019

geometrycollinearincirclemidpoint

Problem Statement

Let

ABC

be a triangle with incenter

I

. The incircle of the triangle is tangent to the sides

BC

and

AC

in points

D

and

E

, respectively. Let

P

denote the common point of lines

AI

and

DE

, and let

M

and

N

denote the midpoints of sides

BC

and

AB

, respectively. Prove that points

M, N

and

P

are collinear.
(Proposed by Karl Czakler)