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.

(A) 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| = 12
    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