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As the flies are on the perimeter of a circle, the quadrilateral

ABCDiscyclic. And there are several properties of cyclic quadrilaterals with sidesa, b, c, dthat we can use to solve this problem.Firstly the

semiperimeterof the quadrilateral is:The area of the quadrilateral is:

And the circumradius,

R, (the radius of the circumcircle) is given by:We know that A, B, C, D lie on a semicircle (as the quadrilateral

ABCDdoesnotcontain the centre of the circumcircle). So as we move from A to B to C to D we are advancing around the semicircle with straight line distancesa, b, cand the final return to A (with distanced) is the longest side of the quadrilateral. So two ofa, b, care the same.Also

dis the longest side, andd < a + b + c. It doesn’t matter what order thea, b, csides occur in, so we will assumea < candb = aorb = c.This Python program considers increasing values for

d, and finds correspondinga, b, cvalues that form a cyclic quadrilateral. It then computes the angles subtended at the centre on the circumcircle by AB, BC and CD, in order for A, B, C, D to lie within a semicircle of the circumcircle these angles must sum to less than 180° (or π radians).The program runs in 53ms.

Solution:The four lengths are: 5, 5, 6, 14 flymins.The radius of the circumcircle is (5/6)√109 (≈ 8.700) flymins, and the flies are contained in a sector of the circumcircle spanning ≈107.13°.

There is another cyclic quadrilateral with integer sides, namely, 2, 5, 5, 8 flymins, but this encloses the centre of the circumcircle as shown below:

In this case the radius of the circumcircle is (5/8)√41 (≈ 4.002) flymins, but the flies span ≈183.6° of the circumcircle.