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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.