[seqfan] Re: Trapezoidal area integers: overlooked types

[Andrew Weimholt]

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