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Problems
Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2022 Sharygin Geometry Olympiad
20
20
Part of
2022 Sharygin Geometry Olympiad
Problems
(1)
Too many perpendicular and parallel lines
Source: Sharygin 2022 P20
3/4/2022
Let
O
O
O
,
I
I
I
be the circumcenter and the incenter of
△
A
B
C
\triangle ABC
△
A
BC
;
R
R
R
,
r
r
r
be the circumradius and the inradius;
D
D
D
be the touching point of the incircle with
B
C
BC
BC
; and
N
N
N
be an arbitrary point of segment
I
D
ID
I
D
. The perpendicular to
I
D
ID
I
D
at
N
N
N
meets the circumcircle of
A
B
C
ABC
A
BC
at points
X
X
X
and
Y
Y
Y
. Let
O
1
O_{1}
O
1
be the circumcircle of
△
X
I
Y
\triangle XIY
△
X
I
Y
. Find the product
O
O
1
⋅
I
N
OO_{1}\cdot IN
O
O
1
⋅
I
N
.
geometry