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All-Russian Olympiad Regional Round
1998 All-Russian Olympiad Regional Round
8.7
8.7
Part of
1998 All-Russian Olympiad Regional Round
Problems
(1)
sum of 3 angles = 180^o, 3 circles - All-Russian MO 1998 Regional (R4) 8.7
Source:
9/17/2024
Let
O
O
O
be the center of a circle circumscribed about an acute angle triangle
A
B
C
ABC
A
BC
,
S
A
S_A
S
A
,
S
B
S_B
S
B
,
S
C
S_C
S
C
- circles with center O, tangent to sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
respectively. Prove that the sum of three angles : between the tangents to
S
A
S_A
S
A
drawn from point
A
A
A
, to
S
B
S_B
S
B
from point
B
B
B
and to
S
C
S_C
S
C
- from point
C
C
C
, is equal to
18
0
o
180^o
18
0
o
.
geometry
angles