**From New Scientist #1508, 15th May 1986** [link]

I call ABCD an odd cyclic quadrilateral, or “odd quad” for short. It has four corners A, B, C, D, and four straight sides AB, BC, CD, DA, so it’s a quadrilateral. The corners lie on a circle, so it’s cyclic. And it’s odd because — well, what is its area? I have decided to define that as what the sides cut off from the outside world, that is, the sum of the shaded areas.

A *neat* odd quad has the lengths of its four sides all different positive whole numbers and its area is a whole number too.

Can you find a neat odd quad with an area less than 30? What are the lengths of:

(a) The sides AB, CD, which don’t cross?

(b) The crossing sides BC, AD?

[enigma359]

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This Python 3 program considers quadrilaterals with increasing perimeter, until it finds one that is a crossed cyclic quadrilateral with an area less than 30. It runs in 214ms.

Solution:Yes, there is a neat odd quad with area less than 30. (a) The two non-crossing sides have length 2 and 6. (b) The two crossing sides have length 7 and 9.AB = 2, BC = 9, CD = 6, AD = 7.

The area of the quadrilateral is 15.

Note that the angles B and D are right angles, so the line AC is a diameter of the circumcircle, and the triangles ABX, CDX are 3:4:5 triangles.

Essentially the same with minor variations.