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National and Regional Contests
Greece Contests
Greece Team Selection Test
2022 Greece Team Selection Test
2
2
Part of
2022 Greece Team Selection Test
Problems
(1)
circumcenter of EFG lies on (ABC)
Source: 2022 Greece TST p2
11/3/2022
Consider triangle
A
B
C
ABC
A
BC
with
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
, inscribed in triangle
Γ
1
\Gamma_1
Γ
1
and the circles
Γ
2
(
B
,
A
C
)
\Gamma_2 (B,AC)
Γ
2
(
B
,
A
C
)
and
Γ
2
(
C
,
A
B
)
\Gamma_2 (C,AB)
Γ
2
(
C
,
A
B
)
. A common point of circle
Γ
2
\Gamma_2
Γ
2
and
Γ
3
\Gamma_3
Γ
3
is point
E
E
E
, a common point of circle
Γ
1
\Gamma_1
Γ
1
and
Γ
3
\Gamma_3
Γ
3
is point
F
F
F
and a common point of circle
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
is point
G
G
G
, where the points
E
,
F
,
G
E,F,G
E
,
F
,
G
lie on the same semiplane defined by line
B
C
BC
BC
, that point
A
A
A
doesn't lie in. Prove that circumcenter of triangle
E
F
G
EFG
EFG
lies on circle
Γ
1
\Gamma_1
Γ
1
.Note: By notation
Γ
(
K
,
R
)
\Gamma (K,R)
Γ
(
K
,
R
)
, we mean random circle
Γ
\Gamma
Γ
has center
K
K
K
and radius
R
R
R
.
geometry
circumcircle