Assume d-c = c-b = b-a = d For Adam's 6 problems the smallest side has following length: y = -2d y = -d x = -4/3d y = 3d x = 0 x = 0 Therefore there is only one family of quadrangles. The one mentioned by Eric. Walter

Date: Sun, 2 Aug 2015 22:53:36 +0200 From: "Adam P. Goucher" <apgoucher@gmx.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Quadrangles Message-ID:

<trinity-6082781c-d98c-470a-8d37-d3953acff651-1438548816659@3c app-mailcom-bs03>

Content-Type: text/plain; charset=UTF-8

Suppose a > b > c > d without loss of generality. If the right-angles were opposite each other, then we would have w^2 + x^2 = y^2 + z^2, where {w,x,y,z} = {a,b,c,d} in some order. It is obvious that the only possibility is a^2 + d^2 = b^2 + c^2 (lest we violate some of the inequalities).

Add to this the assumption that the four sides are in arithmetic progression, we get a + d = b + c, thus a^2 + d^2 + 2ad = b^2 + c^2 + 2bc, thus ad = bc. This implies {a, d} and {b, c} are both valid solution-pairs to some quadratic equation, contradicting the distinct-ness of {a,b,c,d}.

So we can assume the two right-angles are adjacent, and that the four sides are w,x,y,z (with z slanted). Then the relevant equation is:

z^2 = x^2 + (w - y)^2 (assuming wlog w > y)

We also have z > x, obviously, so there are six possibilities for the order (which are six algebraically distinct problems):

z > x > w > y z > w > x > y z > w > y > x w > z > x > y w > z > y > x w > y > z > x

Now, being in an arithmetic progression gives two real constraints, and the aforementioned one is a third constraint, suggesting that the family of solutions has dimension 1. This in turn suggests that up to similarity there are finitely many solutions for each of the six problems.

Sincerely,

Adam P. Goucher

...

Sent: Sunday, August 02, 2015 at 8:11 PM From: "Eric Angelini" <Eric.Angelini@kntv.be> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Quadrangles

Hello MathFunsters, Are there quadrangles having: - two right angles - and their four sides in arithmetic progression? I know only of one family of such quadrangles -- the ones build on the pythagorian triplet 3,4,5. The simplest example has the shape of an U with a tilted roof: - the left vertical wall has size 3, - the horizontal ground has size 4, - the right vertical wall has size 6, - the tilted roof has size 5.

Best, ?.