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From: aegis on 24 Jan 2010 12:41
I bisected the sides of a quadrilateral, and connected the points of
bisection, to produce a geometric object that I want to prove is a
rhomboid. Where any adjacent sides are unequal,
and where the angles formed are not multiples of pi/2.

I did this in a very ugly way by fixing the center of a quadrilateral
at the origin and expressing the vertices of the quadrilateral in
terms
of the points in R^2. I then bisected the sides of the quadrilateral
and connected those points to form a new geometric object.

I derived the points bisected from the quadrilateral and
expressed them as position vectors and then used the position
vectors to derive the sides of the geometric object inscribed
in the quadrilateral.

I represented the sides of the geometric object as vectors
and compared their lengths and angles formed.

This seems lame to me. Is there a more concise
proof using vector geometry?

--
aegis
From: hagman on 24 Jan 2010 12:47
On 24 Jan., 18:41, aegis <ae...(a)mad.scientist.com> wrote:
> I bisected the sides of a quadrilateral, and connected the points of
> bisection, to produce a geometric object that I want to prove is a
> rhomboid.  Where any adjacent sides are unequal,
> and where the angles formed are not multiples of pi/2.
>
> I did this in a very ugly way by fixing the center of a quadrilateral
> at the origin and expressing the vertices of the quadrilateral in
> terms
> of the points in R^2.  I then bisected the sides of the quadrilateral
> and connected those points to form a new geometric object.
>
> I derived the points bisected from the quadrilateral and
> expressed them as position vectors and then used the position
> vectors to derive the sides of the geometric object inscribed
> in the quadrilateral.
>
> I represented the sides of the geometric object as vectors
> and compared their lengths and angles formed.
>
> This seems lame to me.  Is there a more concise
> proof using vector geometry?
>
> --
> aegis

For given A,B,C,D (not necessarily coplanar!)
(B+C)/2 - (A+B)/2 = (C-A)/2 = (C+D)/2 - (A+D)/2
and simlar for the other pair.

More geometrically, the intercept theorem shows immediately
that the sides of your rhomboid are parallel to and half as long as
the diagonals of the given quadrilateral.
From: Ken Pledger on 25 Jan 2010 16:45
In article
<7ba22c6d-6d8e-4d02-9db3-40f68a70920d(a)l19g2000yqb.googlegroups.com>,
aegis <aegis(a)mad.scientist.com> wrote:

> I bisected the sides of a quadrilateral, and connected the points of
> bisection, to produce a geometric object that I want to prove is a
> rhomboid....


Look up "Varignon's Theorem". It works even if the quadrilateral
is skew, i.e. not in a plane.

Ken Pledger.
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