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All-Russian Olympiad Regional Round
2004 All-Russian Olympiad Regional Round
9.2
9.2
Part of
2004 All-Russian Olympiad Regional Round
Problems
(1)
isosceles, medians cut (ABC) - All-Russian MO 2004 Regional (R4) 9.2
Source:
9/27/2024
In triangle
A
B
C
ABC
A
BC
, medians
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
,
C
C
′
CC'
C
C
′
are extended until they intersect with the circumcircle at points
A
0
A_0
A
0
,
B
0
B_0
B
0
,
C
0
C_0
C
0
, respectively. It is known that the intersection point M of the medians of triangle
A
B
C
ABC
A
BC
divides the segment
A
A
0
AA_0
A
A
0
in half. Prove that the triangle
A
0
B
0
C
0
A_0B_0C_0
A
0
B
0
C
0
is isosceles.
geometry
isosceles
Medians