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Baltic Way
2006 Baltic Way
15
15
Part of
2006 Baltic Way
Problems
(1)
The line t through the centroid of triangle ABC
Source: Baltic Way 2006
12/4/2010
Let the medians of the triangle
A
B
C
ABC
A
BC
intersect at point
M
M
M
. A line
t
t
t
through
M
M
M
intersects the circumcircle of
A
B
C
ABC
A
BC
at
X
X
X
and
Y
Y
Y
so that
A
A
A
and
C
C
C
lie on the same side of
t
t
t
. Prove that
B
X
⋅
B
Y
=
A
X
⋅
A
Y
+
C
X
⋅
C
Y
BX\cdot BY=AX\cdot AY+CX\cdot CY
BX
⋅
B
Y
=
A
X
⋅
A
Y
+
CX
⋅
C
Y
.
geometry
circumcircle
trapezoid
geometry proposed