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Greece Contests
Greece Team Selection Test
2015 Greece Team Selection Test
3
3
Part of
2015 Greece Team Selection Test
Problems
(1)
Nice geometry with many circles
Source: Greek BMO TST Problem 3
4/5/2015
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
A
C
<
B
C
\displaystyle{AB<AC<BC}
A
B
<
A
C
<
BC
inscribed in circle
c
(
O
,
R
)
\displaystyle{c(O,R)}
c
(
O
,
R
)
.The excircle
(
c
A
)
\displaystyle{(c_A)}
(
c
A
)
has center
I
\displaystyle{I}
I
and touches the sides
B
C
,
A
C
,
A
B
\displaystyle{BC,AC,AB}
BC
,
A
C
,
A
B
of the triangle
A
B
C
ABC
A
BC
at
D
,
E
,
Z
\displaystyle{D,E,Z}
D
,
E
,
Z
respectively.
A
I
\displaystyle{AI}
A
I
cuts
(
c
)
\displaystyle{(c)}
(
c
)
at point
M
M
M
and the circumcircle
(
c
1
)
\displaystyle{(c_1)}
(
c
1
)
of triangle
A
Z
E
\displaystyle{AZE}
A
ZE
cuts
(
c
)
\displaystyle{(c)}
(
c
)
at
K
K
K
.The circumcircle
(
c
2
)
\displaystyle{(c_2)}
(
c
2
)
of the triangle
O
K
M
\displaystyle{OKM}
O
K
M
cuts
(
c
1
)
\displaystyle{(c_1)}
(
c
1
)
at point
N
N
N
.Prove that the point of intersection of the lines
A
N
,
K
I
AN,KI
A
N
,
K
I
lies on
(
c
)
\displaystyle{(c)}
(
c
)
.
geometry
circumcircle