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National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
1996 Greece Junior Math Olympiad
2
2
Part of
1996 Greece Junior Math Olympiad
Problems
(1)
computational with midpoints and areas (Greece Junior 1996 p2)
Source:
3/16/2020
In a triangle
A
B
C
ABC
A
BC
let
D
,
E
,
Z
,
H
,
G
D,E,Z,H,G
D
,
E
,
Z
,
H
,
G
be the midpoints of
B
C
,
A
D
,
B
D
,
E
D
,
E
Z
BC,AD,BD,ED,EZ
BC
,
A
D
,
B
D
,
E
D
,
EZ
respectively. Let
I
I
I
be the intersection of
B
E
,
A
C
BE,AC
BE
,
A
C
and let
K
K
K
be the intersection of
H
G
,
A
C
HG,AC
H
G
,
A
C
. Prove that: a)
A
K
=
3
C
K
AK=3CK
A
K
=
3
C
K
b)
H
K
=
3
H
G
HK=3HG
HK
=
3
H
G
c)
B
E
=
3
E
I
BE=3EI
BE
=
3
E
I
d)
(
E
G
H
)
=
1
32
(
A
B
C
)
(EGH)=\frac{1}{32}(ABC)
(
EG
H
)
=
32
1
(
A
BC
)
Notation
(
.
.
.
)
(...)
(
...
)
stands for area of
.
.
.
.
....
....
geometry
areas
midpoints