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Problems
Contests
International Contests
Baltic Way
2018 Baltic Way
14
14
Part of
2018 Baltic Way
Problems
(1)
Equal segments in a circumscribed quadrilateral
Source: Baltic Way 2018, Problem 14
11/6/2018
A quadrilateral
A
B
C
D
ABCD
A
BC
D
is circumscribed about a circle
ω
\omega
ω
. The intersection point of
ω
\omega
ω
and the diagonal
A
C
AC
A
C
, closest to
A
A
A
, is
E
E
E
. The point
F
F
F
is diametrally opposite to the point
E
E
E
on the circle
ω
\omega
ω
. The tangent to
ω
\omega
ω
at the point
F
F
F
intersects lines
A
B
AB
A
B
and
B
C
BC
BC
in points
A
1
A_1
A
1
and
C
1
C_1
C
1
, and lines
A
D
AD
A
D
and
C
D
CD
C
D
in points
A
2
A_2
A
2
and
C
2
C_2
C
2
, respectively. Prove that
A
1
C
1
=
A
2
C
2
A_1C_1=A_2C_2
A
1
C
1
=
A
2
C
2
.
geometry