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
1d parallel to 1+d
This is part (A) which I showed is impossible
1d parallel to 12d
Possible if 1/2 < d < 1
(dâ‰ 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 nonparallel sides intersect each other.
1 parallel to 1d
Not possible. Requires x=1+d which makes the points collinear.
1 parallel to 1+2d
Not possible. The two nonparallel sides intersect each other.

Posted by Jer
on 20150420 12:04:03 