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median concurrency with equal lengths

Source: Sharygin 2023 - P15 (Grade-9-10)

March 4, 2023

geometryconcurrencySharygin Geometry OlympiadSharygin 2023

Problem Statement

Let

ABCD

be a convex quadrilateral. Points

X

and

Y

lie on the extensions beyond

D

of the sides

CD

and

AD

respectively in such a way that

DX = AB

and

DY = BC

. Similarly points

Z

and

T

lie on the extensions beyond

B

of the sides

CB

and

AB

respectively in such a way that

BZ = AD

and

BT = DC

. Let

M_1

be the midpoint of

XY

, and

M_2

be the midpoint of

ZT

. Prove that the lines

DM_1, BM_2

and

AC

concur.

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