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Greece Contests
Greece Team Selection Test
2017 Greece Team Selection Test
1
1
Part of
2017 Greece Team Selection Test
Problems
(1)
Geometry: Concurrent lines and concyclic points.
Source: Greece team selection test problem 1
5/3/2017
Let
A
B
C
ABC
A
BC
be an acute-angled triangle inscribed in circle
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
with
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
, and
c
1
c_1
c
1
be the inscribed circle of
A
B
C
ABC
A
BC
which intersects
A
B
,
A
C
,
B
C
AB, AC, BC
A
B
,
A
C
,
BC
at
F
,
E
,
D
F, E, D
F
,
E
,
D
respectivelly. Let
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
be points which lie on
c
c
c
such that the quadrilaterals
A
E
F
A
′
,
B
D
F
B
′
,
C
D
E
C
′
AEFA', BDFB', CDEC'
A
EF
A
′
,
B
D
F
B
′
,
C
D
E
C
′
are inscribable. (1) Prove that
D
E
A
′
B
′
DEA'B'
D
E
A
′
B
′
is inscribable. (2) Prove that
D
A
′
,
E
B
′
,
F
C
′
DA', EB', FC'
D
A
′
,
E
B
′
,
F
C
′
are concurrent.
geometry