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Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1975 Vietnam National Olympiad
3
3
Part of
1975 Vietnam National Olympiad
Problems
(1)
vol KOAC/vol KOBD = AC/BD iff 2AC·BD = AB^2 in a tetrahedron ABCD
Source: Vietnamese MO (VMO) 1975 P3
8/20/2018
Let
A
B
C
D
ABCD
A
BC
D
be a tetrahedron with
B
A
⊥
A
C
,
D
B
⊥
(
B
A
C
)
BA \perp AC,DB \perp (BAC)
B
A
⊥
A
C
,
D
B
⊥
(
B
A
C
)
. Denote by
O
O
O
the midpoint of
A
B
AB
A
B
, and
K
K
K
the foot of the perpendicular from
O
O
O
to
D
C
DC
D
C
. Suppose that
A
C
=
B
D
AC = BD
A
C
=
B
D
. Prove that
V
K
O
A
C
V
K
O
B
D
=
A
C
B
D
\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD}
V
K
OB
D
V
K
O
A
C
=
B
D
A
C
if and only if
2
A
C
⋅
B
D
=
A
B
2
2AC \cdot BD = AB^2
2
A
C
⋅
B
D
=
A
B
2
.
geometry
3D geometry
tetrahedron