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National and Regional Contests
Ecuador Contests
Ecuador Mathematical Olympiad (OMEC)
2020 Ecuador NMO (OMEC)
3
3
Part of
2020 Ecuador NMO (OMEC)
Problems
(1)
Finding two circumcenters
Source: OMEC Ecuador National Olympiad Final Round 2020 N3 P3 day 1
11/10/2024
Let
A
B
C
ABC
A
BC
a triangle with circumcircle
Γ
\Gamma
Γ
and circumcenter
O
O
O
. A point
X
X
X
, different from
A
A
A
,
B
B
B
,
C
C
C
, or their diametrically opposite points, on
Γ
\Gamma
Γ
, is chosen. Let
ω
\omega
ω
the circumcircle of
C
O
X
COX
COX
. Let
E
E
E
the second intersection of
X
A
XA
X
A
with
ω
\omega
ω
,
F
F
F
the second intersection of
X
B
XB
XB
with
ω
\omega
ω
and
D
D
D
a point on line
A
B
AB
A
B
such that
C
D
⊥
E
F
CD \perp EF
C
D
⊥
EF
. Prove that
E
E
E
is the circumcenter of
A
D
C
ADC
A
D
C
and
F
F
F
is the circumcenter of
B
D
C
BDC
B
D
C
.
geometry
circumcircle