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National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2019 Sharygin Geometry Olympiad
15
15
Part of
2019 Sharygin Geometry Olympiad
Problems
(1)
Sharygin CR 2019 P15
Source: Sharygin CR 2019 P15 (Grade 9 - 11)
3/6/2019
The incircle
ω
\omega
ω
of triangle
A
B
C
ABC
A
BC
touches the sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
at points
D
D
D
,
E
E
E
and
F
F
F
respectively. The perpendicular from
E
E
E
to
D
F
DF
D
F
meets
B
C
BC
BC
at point
X
X
X
, and the perpendicular from
F
F
F
to
D
E
DE
D
E
meets
B
C
BC
BC
at point
Y
Y
Y
. The segment
A
D
AD
A
D
meets
ω
\omega
ω
for the second time at point
Z
Z
Z
. Prove that the circumcircle of the triangle
X
Y
Z
XYZ
X
Y
Z
touches
ω
\omega
ω
.
geometry