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.

Using similar reasoning as I did for part (A) but checking 5 other possibilities for which sides are parallel yields the following

1-d parallel to 1+d
This is part (A) which I showed is impossible

1-d parallel to 1-2d
Possible if -1/2 < d < 1
(d≠0 otherwise PS parallel to QR)

1+d parallel to 1+2d
Not possible.  Requires x=1, similar to part (A).

1 parallel to 1+d
Not possible.  The two non-parallel sides intersect each other.

1 parallel to 1-d
Not possible.  Requires x=1+d which makes the points collinear.

1 parallel to 1+2d
Not possible.  The two non-parallel sides intersect each other.

Posted by Jer on 2015-04-20 12:04:03