[seqfan] Re: Trapezoidal area integers: overlooked types

[Charles Greathouse]

> If "trapezoid" is defined as a quadrilateral with "at least one" pair of
> parallel edges, then squares, rectangles, and parallelograms are also
> trapezoids, in which case every positive integer should be included.
> (a 1 by L rectangle has integer sides: 1,L,1,L, and integer area L)

I had looked into it for that reason: to discover which definition was used.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Mon, Aug 6, 2012 at 4:04 AM, Andrew Weimholt
<andrew.weimholt at gmail.com> wrote:
> On Sat, Aug 4, 2012 at 1:33 PM, Alonso Del Arte
> <alonso.delarte at gmail.com> wrote:
>>
>> Have I overlooked any other kinds of trapezoids than can have integers for
>> area and all four sides? And if so, does this mean that almost all
>> composite numbers are in A214602?
>>
>
> If "trapezoid" is defined as a quadrilateral with "at least one" pair of
> parallel edges, then squares, rectangles, and parallelograms are also
> trapezoids, in which case every positive integer should be included.
> (a 1 by L rectangle has integer sides: 1,L,1,L, and integer area L)
>
> However, if we define "trapezoid" as a quadrilateral with "exactly one"
> pair of parallel edges, then there are still some that you missed...
>
> It is not true that after removing the largest rectangle, the two
> remaining pythagorean triangles must have the same length for their
> smallest side. Take the pythagorean triangles (6,8,10) and (8,15,17).
> We can put an 8 x L rectangle in between them, and we get a rectangle
> with area 84+8L.
>
> It is also a faulty assumption that the non parallel sides must have
> slopes with opposite signs with respect to the base. We can attach
> the (6,8,10) and (8,15,17) directly to each other with no rectangle in
> between, and still get a trapezoid (with area 84) by flipping one of
> the triangles vertically (sides are 6,10,15,17).  Furthermore, we
> can remove anywhere from 1 to 5 units of length from the parallel
> edges from this trapezoid and still get a trapezoid with unit area.
> We are merely removing parallelograms with unit area instead of
> rectangles. If we remove 5 units, we get a trapezoid with area 44,
> and sides (1,10,10,17)
>
> Andrew
>
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