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angle bisector <BDC$ // AS, angle bisector of <BAC, midpoints, // related

Source: 2014 Croatia MO p3

August 5, 2020

geometryangle bisectorparallel

Problem Statement

Given a triangle

ABC

in which

|AB|>|AC|

. Let

P

be the midpoint of the side

BC

, and

S

the point in which the angle bisector of

\angle BAC

intersects that side. The parallel with the line

AS

through the point

P

intersects lines

AB

and

AC

at points

X

and

Y

respectively . Let

Z

be the point be such that

Y

is the midpoint of the length

XZ

and let the lines

BY

and

CZ

intersect at point

D

. Prove that the angle bisector of

\angle BDC

is parallel to the lines

AS

.

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