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National and Regional Contests
Azerbaijan Contests
Azerbaijan IZHO TST
2020 Azerbaijan IZHO TST
2
2
Part of
2020 Azerbaijan IZHO TST
Problems
(1)
Geometry Problem
Source: Azerbaijan Math Olympiad Training
12/15/2019
Consider two circles
k
1
,
k
2
k_1,k_2
k
1
,
k
2
touching at point
T
T
T
. A line touches
k
2
k_2
k
2
at point
X
X
X
and intersects
k
1
k_1
k
1
at points
A
,
B
A,B
A
,
B
where
B
B
B
lies between
A
A
A
and
X
X
X
.Let
S
S
S
be the second intersection point of
k
1
k_1
k
1
with
X
T
XT
XT
. On the arc \overarc{TS} not containing
A
A
A
and
B
B
B
, a point
C
C
C
is choosen. Let
C
Y
CY
C
Y
be the tangent line to
k
2
k_2
k
2
with
Y
∈
k
2
Y\in{k_2}
Y
∈
k
2
, such that the segment
C
Y
CY
C
Y
doesn't intersect the segment
S
T
ST
ST
.If
I
=
X
Y
∩
S
C
I=XY\cap{SC}
I
=
X
Y
∩
SC
, prove that :
(
a
)
(a)
(
a
)
the points
C
,
T
,
Y
,
I
C,T,Y,I
C
,
T
,
Y
,
I
are concyclic.
(
b
)
(b)
(
b
)
I
I
I
is the
A
−
e
x
c
e
n
t
e
r
A-excenter
A
−
e
x
ce
n
t
er
of
△
A
B
C
\triangle ABC
△
A
BC
geometry
TST