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|Arithmetic Quadrilateral (Posted on 2015-04-19)
PQRS is a convex quadrilateral where PQ parallel to RS but PS is not parallel to QR.
(A) Prove that the lengths of its sides PQ, QR, RS and SP (in this order) do not form an arithmetic sequence.
(B) Does there exist such a quadrilateral for which the lengths of its sides PQ, QR, RS and SP form an arithmetic sequence after the order of the lengths in (A) is changed?
If so, provide at least an example with proof.
If not, prove the nonexistance of such a quadrilateral.
Comment 4 of 4 |
If |PQ|, |QR|, |RS|, |SP| are in arithmetic progression
then |QR| + |RS| = |SP| + |PQ|
which proves that
P and R lie on an ellipse with focal
points at S and Q.
If PQ and RS are parallel, then
they form equal angles
with SQ and have point symmetry
about the centre, O,
of the ellipse. Thus PR is a
diameter and the diagonals
bisect each other showing that PQRS
is a parallelogram.
This contradicts the given conditions, so the quadrilateral
as prescribed cannot have sides in arithmetic progression.
(B) If the smallest and largest sides are parallel then the
required trapezium is possible. For example:
|AB| = 3, |AD| = 6, |BC| = 9, |CD| =
is a trapezium with base DC parallel
to AB and with
height 4*sqrt2, made up of a
rectangle (3 x 4sqrt2),
supported by end triangles with base
lengths 2 and 7.
Posted by Harry
on 2015-04-20 19:04:13
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