All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 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.

 No Solution Yet Submitted by K Sengupta No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 example | Comment 3 of 4 |
WLOG Let PQ=1-d, QR=1, RS=1+2d, SP=1+d.

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+2d,0)

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

[(x+1+d)-(1+2d)]²+[√(1-x²)]²=[1+d]²
x²-6xd+9d²+1-x²=1+2d+d²
x=(4d-1)/3

We require -1<x<1
so -1<(4d-1)/3<1
-1/2<d<1
(But d≠0 or we get a rhombus)

Example: d=1/2
x=1/3
P=(5/6,2√(3)/3)
Q=(1/3,2√(3)/3)
R=(0,0)
S=(2,0)

PQ=1/2
QR=1
PS=2
SP=3/2
and PQ parallel to RS

 Posted by Jer on 2015-04-20 12:14:45
Please log in:
 Login: Password: Remember me: Sign up! | Forgot password

 Search: Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (1)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2018 by Animus Pactum Consulting. All rights reserved. Privacy Information