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National and Regional Contests
North Macedonia Contests
JBMO TST - Macedonia
2017 Macedonia JBMO TST
2
2
Part of
2017 Macedonia JBMO TST
Problems
(1)
Rectangle and medians [Macedonia JBMO TST 2017]
Source: Macedonia JBMO TST 2017, Problem 2
6/26/2018
In the triangle
A
B
C
ABC
A
BC
, the medians
A
A
1
AA_1
A
A
1
,
B
B
1
BB_1
B
B
1
, and
C
C
1
CC_1
C
C
1
are concurrent at a point
T
T
T
such that
B
A
1
=
T
A
1
BA_1=TA_1
B
A
1
=
T
A
1
. The points
C
2
C_2
C
2
and
B
2
B_2
B
2
are chosen on the extensions of
C
C
1
CC_1
C
C
1
and
B
B
2
BB_2
B
B
2
, respectively, such that C_1C_2 = \frac{CC_1}{3} \text{and} B_1B_2 = \frac{BB_1}{3}. Show that
T
B
2
A
C
2
TB_2AC_2
T
B
2
A
C
2
is a rectangle.
geometry
rectangle