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International Contests
IMO Longlists
1970 IMO Longlists
7
7
Part of
1970 IMO Longlists
Problems
(1)
Squares Constructed on Sides of Quadrilateral
Source: ILL 1970 - Problem 7.
5/24/2011
Let
A
B
C
D
ABCD
A
BC
D
be an arbitrary quadrilateral. Squares with centers
M
1
,
M
2
,
M
3
,
M
4
M_1, M_2, M_3, M_4
M
1
,
M
2
,
M
3
,
M
4
are constructed on
A
B
,
B
C
,
C
D
,
D
A
AB,BC,CD,DA
A
B
,
BC
,
C
D
,
D
A
respectively, all outwards or all inwards. Prove that
M
1
M
3
=
M
2
M
4
M_1 M_3=M_2 M_4
M
1
M
3
=
M
2
M
4
and
M
1
M
3
⊥
M
2
M
4
M_1 M_3\perp M_2 M_4
M
1
M
3
⊥
M
2
M
4
.
geometry
geometry proposed