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National and Regional Contests
Mexico Contests
Mexico National Olympiad
1996 Mexico National Olympiad
1
1
Part of
1996 Mexico National Olympiad
Problems
(1)
parallelogram criterion with trisection of a diagonal
Source: Mexican Mathematical Olympiad 1996 OMM P1
7/28/2018
Let
P
P
P
and
Q
Q
Q
be the points on the diagonal
B
D
BD
B
D
of a quadrilateral
A
B
C
D
ABCD
A
BC
D
such that
B
P
=
P
Q
=
Q
D
BP = PQ = QD
BP
=
PQ
=
Q
D
. Let
A
P
AP
A
P
and
B
C
BC
BC
meet at
E
E
E
, and let
A
Q
AQ
A
Q
meet
D
C
DC
D
C
at
F
F
F
. (a) Prove that if
A
B
C
D
ABCD
A
BC
D
is a parallelogram, then
E
E
E
and
F
F
F
are the midpoints of the corresponding sides. (b) Prove the converse of (a).
geometry
parallelogram