perplexus.info :: Geometry : Parallelogram Requirement

Parallelogram Requirement (Posted on 2013-07-07)

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.

Submitted by Bractals

Rating: 3.0000 (1 votes)

Solution: (Hide)

Let PQ denote the vector from point P to point Q.
Let PQ·RS denote the dot product of vector PQ and RS
with PQ·PQ = |PQ|² = |PQ|².

|AB|² + |BC|² + |CD|² + |DA|² =

  AB·AB + BC·BC + CD·CD + DA·DA =

  AB·AB + BC·BC + CD·CD + DA·DA +
    2AB·(AB + BC + CD + DA) =

  AB·AB + 2AB·BC + BC·BC +
    AB·AB + 2AB·DA + DA·DA +
    AB·AB + 2AB·CD + CD·CD =

  AB·AB + 2AB·BC + BC·BC +
    BA·BA + 2BA·AD + AD·AD +
    AB·AB + 2AB·CD + CD·CD =

  (AB + BC)·(AB + BC) +
    (BA + AD)·(BA + AD) +
    (AB + CD)·(AB + CD) =

  AC·AC + BD·BD +
    (AB + CD)·(AB + CD) =

|AC|² + |BD|² + |AB + CD|²

Therefore,

|AB|² + |BC|² + |CD|² + |DA|² = |AC|² + |BD|² ⇔

  |AB + CD|² = 0 ⇔ AB + CD = 0 ⇔ AB = DC ⇔

ABCD is a parallelogram.

QED

Comments: ( You must be logged in to post comments.)

	Subject	Author	Date
	Proof. No picture.	Jer	2013-07-09 11:00:04

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