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.

WLOG Let PQ=1-d, QR=1, RS=1+d, SP=1+2d.

Assume PQRS is a convex quadrilateral where PQ parallel to RS but PS is not parallel to QR.

Let
P=(x+1-d,√(1-x²))
Q=(x,√(1-x²))
R=(0,0)
S=(1+d,0)

The first three distances are clearly true so we need only to solve for SP.

[(x+1+d)-(1+d)]²+[√(1-x²)]²=[1+2d]²
x²-4xd+4d²+1-x²=1+4d+4d²
x=(4d)/(-4d)=-1

Which puts all the points on the x-axis.  So the points are collinear and do not form a quadrilateral.  (Also PS is parallel to QR.)

Posted by Jer on 2015-04-20 10:55:27