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MathLinks Contest 4th
1.3
1.3
Part of
MathLinks Contest 4th
Problems
(1)
0413 geometry 4th edition Round 1 p3
Source:
5/7/2021
Let
Ω
1
(
O
1
,
r
1
)
\Omega_1(O_1, r_1)
Ω
1
(
O
1
,
r
1
)
and
Ω
2
(
O
2
,
r
2
)
\Omega_2(O_2, r_2)
Ω
2
(
O
2
,
r
2
)
be two circles that intersect in two points
X
,
Y
X, Y
X
,
Y
. Let
A
,
C
A, C
A
,
C
be the points in which the line
O
1
O
2
O_1O_2
O
1
O
2
cuts the circle
Ω
1
\Omega_1
Ω
1
, and let
B
B
B
be the point in which the circle
Ω
2
\Omega_2
Ω
2
itnersect the interior of the segment
A
C
AC
A
C
, and let
M
M
M
be the intersection of the lines
A
X
AX
A
X
and
B
Y
BY
B
Y
. Prove that
M
M
M
is the midpoint of the segment
A
X
AX
A
X
if and only if
O
1
O
2
=
1
2
(
r
1
+
r
2
)
O_1O_2 =\frac12 (r_1 + r_2)
O
1
O
2
=
2
1
(
r
1
+
r
2
)
.
geometry
4th edition