MathDB
Problems
Contests
International Contests
Austrian-Polish
1979 Austrian-Polish Competition
8
8
Part of
1979 Austrian-Polish Competition
Problems
(1)
AB^2+BC^2+CD^2+DA^2 = AC^2+BD^2+4MN^2 in 3D, with midpoints
Source: Austrian-Polish 1979
5/25/2019
Let
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
be four points in space, and
M
M
M
and
N
N
N
be the midpoints of
A
C
AC
A
C
and
B
D
BD
B
D
, respectively. Show that
A
B
2
+
B
C
2
+
C
D
2
+
D
A
2
=
A
C
2
+
B
D
2
+
4
M
N
2
AB^2+BC^2+CD^2+DA^2 = AC^2+BD^2+4MN^2
A
B
2
+
B
C
2
+
C
D
2
+
D
A
2
=
A
C
2
+
B
D
2
+
4
M
N
2
geometry
3D geometry
midpoints