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National and Regional Contests
Mathlinks Contests.
MathLinks Contest 2nd
1.3
1.3
Part of
MathLinks Contest 2nd
Problems
(1)
0213 geometry 2nd edition Round 1 p3
Source:
5/10/2021
Given are on a line three points
A
,
B
,
C
A, B, C
A
,
B
,
C
such that
A
B
=
1
AB = 1
A
B
=
1
and
B
C
=
x
BC = x
BC
=
x
. Consider the circles
Ω
a
,
Ω
b
\Omega_a, \Omega_b
Ω
a
,
Ω
b
and
Ω
c
\Omega_c
Ω
c
which are tangent to the given line at the points
A
,
B
,
C
A, B, C
A
,
B
,
C
respectively, and such that
Ω
b
\Omega_b
Ω
b
is tangent externally with both
Ω
a
\Omega_a
Ω
a
and
Ω
c
\Omega_c
Ω
c
in points
M
,
N
M, N
M
,
N
respectively. Find all values of the radius of the circle
Ω
b
\Omega_b
Ω
b
for which the triangle
B
M
N
BMN
BMN
is isosceles.
geometry
2nd edition