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points X', Y' ,Z' coincide wanted, D'X'= AX, E'Y' = BY, F'Z' = CZ, incircle

Source: 2015 XVIII All-Ukrainian Tournament of Young Mathematicians named after M. Y. Yadrenko, Qualifying p24

May 7, 2021

geometryequal segmentsUkrainian TYM

Problem Statement

The inscribed circle

\omega

of the triangle

ABC

touches its sides

BC, CA

, and

AB

at the points

D, E

, and

F

, respectively. Let the points

X, Y

, and

Z

of the circle

\omega

be diametrically opposite to the points

D, E

, and

F

, respectively. Line

AX, BY

and

CZ

intersect the sides

BC, CA

and

AB

at the points

D', E'

and

F'

, respectively. On the segments

AD', BE'

and

CF'

noted the points

X', Y'

and

Z'

, respectively, so that

F'Z' = CZ

,

E'Y' = BY

,

F'Z' = CZ

. Prove that the points

X', Y'

and

Z'

coincide.

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