CoCalc Shared Filessagelets / Question.html
Author: Carl Eberhart
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Sage Cell Server

#### Partial answer to Question arising from Problem 4 plus more questions.

Problem: Suppose Q is a (convex) quadliateral with sides 5, 7, 8 and 9. Suppose also that the spread of the diagonals is 3/4. Find the diagonals of the quadrilateral. Generalize.

Question: Find the quadrilaterals with rational diagonal spread whose sides have integer length.
There are some easy examples -- Any kite with sides of integer length (in fact, any kite ) has a spread of 1. Also, any quadrilateral with integer sides can be flexed to an infinite number of positions so that the diagonal spread is rational, although in general it is difficult to see how to obtain even one position for a random quadrilateral.

We can form a quadrilateral from a single Pythagorean triangle (a right triangle whose sides have integer length $a,b$, and whose hypotenuse has integer length $c$) in several ways. Here are some.
Three ways involve taking two copies of the triangle: 1) Join them at the hypotenuse to form an $a$ by $b$ rectangle. 2) Also join them on side $a$ to form a $c$ by $b$ parallelogram or 3) on side $b$ to form a $c$ by $a$ parallelogram.
Another three ways: 1) Take an integer $n\gt a$ and form a rectangle $b$ by $n$. Now cut off a triangular corner: $b$ by $a$, forming a trapezoid with sides of length $n$, $b$, $n-a$, and $c$. 2) Form a rectangle $2\,b$ by $3\,a$ and from one end cut off a $2\,b$ by $2\,a$ corner. Then from the other end cut off two $a$ by $b$ corners. What's left will be a trapezoid with sides $2\,a$, $c$, $c$, and $2\,c$. Pythagorean triples ($a,\,b,\,c$) are generated by picking random positive integers $x,y$ with $x\lt y$ and forming $a=2\,x\,y$, $b=y^2-x^2$, and $c=x^2+y^2$. Below is a Sagelet which can be used to generate quadrilaterals in this class.
Related Question. Given a positive rational number $r\lt 1$, find a quadrilateral with integer sides and diagonal spread $r$.