**From New Scientist #1500, 20th March 1986** [link]

Four unfriendly flies are sitting watching the cricket. Each is on the boundary of the ground, which is perfectly circular. Their names, in order round the edge, are At, Bet, Cot and Dut.

The length of each of the straight lines At-Bet, Bet-Cot, Cot-Dut and Dut-At is an exact whole number of flymins (this is, the flies’ unit of distance). Two of those lines actually have the same length. Furthermore, the total of these four lengths is precisely the same number as the area (in square flymins) of the quadrilateral At-Bet-Cot-Dut.

If I tell you that that area does not include the pitch (which is at the centre of the ground), can you tell me the four lengths?

[enigma351]

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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.