Because the value can be calculated, it means that no matter how the three circles are placed, the value will not change. In this way, the position of the circle will be adjusted, so that the endpoint of the arc and the center of the corresponding arc are connected in turn, and finally a regular hexagon can be obtained. The internal angle of a regular hexagon is 120, that is, α 1, α2 and α3 are 120. The substitution formula is as follows.

Cos40 cos80-sin40 sin80, sum and difference of products sin α sin β = [cos (α-β)-cos (α+β)]/2cosα cos β = [cos (α+β)+cos (α-β)]/2.

(cos 120+cos 40)/2-(cos 40-cos 120)/2 = cos 120 =- 1/2