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All-Russian Olympiad Regional Round
2000 All-Russian Olympiad Regional Round
9.4
9.4
Part of
2000 All-Russian Olympiad Regional Round
Problems
(1)
concyclic with fixed center if AB = DE - All-Russian MO 2000 Regional (R4) 9.4
Source:
9/26/2024
Circles
S
1
S_1
S
1
and
S
2
S_2
S
2
intersect at points
M
M
M
and
N
N
N
. Through point
A
A
A
of circle
S
1
S_1
S
1
, draw straight lines
A
M
AM
A
M
and
A
N
AN
A
N
intersecting
S
2
S_2
S
2
at points
B
B
B
and
C
C
C
, and through point
D
D
D
of circle
S
2
S_2
S
2
, draw straight lines
D
M
DM
D
M
and
D
N
DN
D
N
intersecting
S
1
S_1
S
1
at points
E
E
E
and
F
F
F
, and
A
A
A
,
E
E
E
,
F
F
F
lie along one side of line
M
N
MN
MN
, and
D
D
D
,
B
B
B
,
C
C
C
lie on the other side (see figure). Prove that if
A
B
=
D
E
AB = DE
A
B
=
D
E
, then points
A
A
A
,
F
F
F
,
C
C
C
and
D
D
D
lie on the same circle, the position of the center of which does not depend on choosing points
A
A
A
and
D
D
D
. https://cdn.artofproblemsolving.com/attachments/7/0/d1f9c2f39352e2b39e55bd2538677073618ef9.png
geometry
circles
Concyclic