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Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine National Mathematical Olympiad
2021 Ukraine National Mathematical Olympiad
10.4
10.4
Part of
2021 Ukraine National Mathematical Olympiad
Problems
(1)
circumcenter of OIH lies on line DE if OI// BC, incenter, orthocenter
Source: 2021 Ukraine NMO 10.4 11.3
4/2/2021
Let
O
,
I
,
H
O, I, H
O
,
I
,
H
be the circumcenter, the incenter, and the orthocenter of
△
A
B
C
\triangle ABC
△
A
BC
. The lines
A
I
AI
A
I
and
A
H
AH
A
H
intersect the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
for the second time at
D
D
D
and
E
E
E
, respectively. Prove that if
O
I
∥
B
C
OI \parallel BC
O
I
∥
BC
, then the circumcenter of
△
O
I
H
\triangle OIH
△
O
I
H
lies on
D
E
DE
D
E
.(Fedir Yudin)
geometry
Circumcenter