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Adygea Teachers' Geometry Olympiad
2021 Adygea Teachers' Geometry Olympiad
2021 Adygea Teachers' Geometry Olympiad
Part of
Adygea Teachers' Geometry Olympiad
Subcontests
(4)
4
1
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4 spheres, each tangent to other 3 (2021 Adygea Teachers' Geometry Olympiad p4)
Two identical balls of radius
15
\sqrt{15}
15
and two identical balls of a smaller radius are located on a plane so that each ball touches the other three. Find the area of the surface
S
S
S
of the ball with the smaller radius.
3
1
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equal segments by excircles (2021 Adygea Teachers' Geometry Olympiad p3)
In a triangle, one excircle touches side
A
B
AB
A
B
at point
C
1
C_1
C
1
and the other touches side
B
C
BC
BC
at point
A
1
A_1
A
1
. Prove that on the straight line
A
1
C
1
A_1C_1
A
1
C
1
the constructed excircles cut out equal segments.
2
1
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isosceles ?, incircle excirle (2021 Adygea Teachers' Geometry Olympiad p2)
In triangle
A
B
C
ABC
A
BC
, the incircle touches the side
A
C
AC
A
C
at point
B
1
B_1
B
1
and one excircle is touching the same side at point
B
2
B_2
B
2
. It is known that the segments
B
B
1
BB_1
B
B
1
and
B
B
2
BB_2
B
B
2
are equal. Is it true that
△
A
B
C
\vartriangle ABC
△
A
BC
is isosceles?
1
1
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tangent segment of 2 circles (2021 Adygea Teachers' Geometry Olympiad p1)
a) Two circles of radii
6
6
6
and
24
24
24
are tangent externally. Line
ℓ
\ell
ℓ
touches the first circle at point
A
A
A
, and the second at point
B
B
B
. Find
A
B
AB
A
B
. b) The distance between the centers
O
1
O_1
O
1
and
O
2
O_2
O
2
of circles of radii
6
6
6
and
24
24
24
is
36
36
36
. Line
ℓ
\ell
ℓ
touches the first circle at point
A
A
A
, and the second at point
B
B
B
and intersects
O
1
O
2
O_1O_2
O
1
O
2
. Find
A
B
AB
A
B
.