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Parallelogram Requirement (Posted on 2013-07-07) Difficulty: 3 of 5

Let ABCD be a convex quadrilateral.

Prove that ABCD is a parallelogram
if and only if
|AC|2 + |BD|2 = |AB|2 + |BC|2 + |CD|2 + |DA|2.

See The Solution Submitted by Bractals    
Rating: 3.0000 (1 votes)

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Solution Proof. No picture. Comment 1 of 1
Start with the "if":

Call the four vertices of the parallelogram:
A=(0,0) B=(a,0) C=(a+b,c) D=(b,c)
|AC|2 + |BD|2
=((a+b)+c)+((b-a)+c
=2a+2b+2c

|AB|2 + |BC|2 + |CD|2 + |DA|2
=2(b+c)+2(a)
=2a+2b+2c

For the "only if" I'll show if the equation works for a trapezoid, the trapezoid must be a parallelogram.
A=(0,0) B=(a,0) C=((b+x),c) D=(b,c)
|AC|2 + |BD|2 = |AB|2 + |BC|2 + |CD|2 + |DA|2
((b+x)+c)+((b-a)+c) = (b+c)+x+((b+x-a)+c)+a
x-2ax+a=0
(x-a)=0
x=a

What I'd like to see is a visual proof such as the three on the wikipedia page
http://en.wikipedia.org/wiki/Pythagorean_theorem


  Posted by Jer on 2013-07-09 11:00:04
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