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1976 All Soviet Union Mathematical Olympiad
222
222
Part of
1976 All Soviet Union Mathematical Olympiad
Problems
(1)
ASU 222 All Soviet Union MO 1976 sum of arcs =180^o , 3 equal circles
Source:
7/6/2019
Given three circumferences of the same radius in a plane. a) All three are crossing in one point
K
K
K
. Consider three arcs
A
K
,
C
K
,
E
K
AK,CK,EK
A
K
,
C
K
,
E
K
: the
A
,
C
,
E
A,C,E
A
,
C
,
E
are the points of the circumferences intersection and the arcs are taken in the clockwise direction. Every arc is inside one circle, outside the second and on the border of the third one. Prove that the sum of the arcs is
180
180
180
degrees. b) Consider the case, when the three circles give a curvilinear triangle
B
D
F
BDF
B
D
F
as their intersection (instead of one point
K
K
K
). The arcs are taken in the clockwise direction. Every arc is inside one circle, outside the second and on the border of the third one. Prove that the sum of the
A
B
,
C
D
AB, CD
A
B
,
C
D
and
E
F
EF
EF
arcs is
180
180
180
degrees.
geometry
equal circles
arc