MathDB
Problems
Contests
National and Regional Contests
Greece Contests
Greece JBMO TST
2001 Greece JBMO TST
2
2
Part of
2001 Greece JBMO TST
Problems
(1)
computational in ABCD, angles and midpoints related, area wanted
Source: Greece JBMO TST 2001 p2
6/17/2019
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral with
∠
D
A
B
=
6
0
o
\angle DAB=60^o
∠
D
A
B
=
6
0
o
,
∠
A
B
C
=
6
0
o
\angle ABC=60^o
∠
A
BC
=
6
0
o
and
∠
B
C
D
=
12
0
o
\angle BCD=120^o
∠
BC
D
=
12
0
o
. Diagonals
A
C
AC
A
C
,
B
D
BD
B
D
intersect at point
M
M
M
and
B
M
=
a
,
M
D
=
2
a
BM=a, MD=2a
BM
=
a
,
M
D
=
2
a
. Let
O
O
O
be the midpoint of side
A
C
AC
A
C
and draw
O
H
⊥
B
D
,
H
∈
B
D
OH \perp BD, H \in BD
O
H
⊥
B
D
,
H
∈
B
D
and
M
N
⊥
O
B
,
N
∈
O
B
MN\perp OB, N \in OB
MN
⊥
OB
,
N
∈
OB
. Prove that i)
H
M
=
M
N
=
a
2
HM=MN=\frac{a}{2}
H
M
=
MN
=
2
a
ii)
A
D
=
D
C
AD=DC
A
D
=
D
C
iii)
S
A
B
C
D
=
9
a
2
2
S_{ABCD}=\frac{9a^2}{2}
S
A
BC
D
=
2
9
a
2
geometry
area