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Kvant 2024
M2804
M2804
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Kvant 2024
Problems
(1)
Weird geo with two equal circles and centroid
Source: Caucasus MO 2024, Juniors P8
3/15/2024
There are two equal circles of radius
1
1
1
placed inside the triangle
A
B
C
ABC
A
BC
with side
B
C
=
6
BC = 6
BC
=
6
. The circles are tangent to each other, one is inscribed in angle
B
B
B
, the other one is inscribed in angle
C
C
C
. (a) Prove that the centroid
M
M
M
of the triangle
A
B
C
ABC
A
BC
does not lie inside any of the given circles. (b) Prove that if
M
M
M
lies on one of the circles, then the triangle
A
B
C
ABC
A
BC
is isosceles.
geometry