>>14079041Programmer here.
The triangle between their centres is (10,14,16). The angle at the centre of circle of diameter 10 is acos(1/7), of the one of 6 is Pi/3, and of 4 is acos(11/14).
The length-14 side must be rotated such that one end is 6 units above the other one, so the angle by which the configuration is rotated is -asin(3/7), and the horizontal distance between them is sqrt(7^2-3^2)=4*sqrt(10).
The angle from the small one to the medium one is then acos(11/14)+asin(6/14) clockwise from lefttwards, and that from the large to the medium one is pi-(acos(1/7)-asin(6/14)).
So the coordinates on the edges of the overlaid triangle are (0,0), (4*sqrt(10),0) and (cos(pi-(acos(1/7)-asin(6/14)))*8,5+3+sin(pi-(acos(1/7)-asin(6/14)))*8).
The angle from the origin to the top of the medium one is then 2*Pi-(atan(y3/x3)+Pi*(x3<0?=1)+atan(y3/(x2-x3))+Pi*((x2-x3)<0?=0))=Pi-(atan(y3/x3)+atan(y3/(x2-x3))).
Inputting this to WolframAlpha with values substituted in: -((-atan((5+3+sin(pi-(acos(1/7)-asin(6/14)))*8)/(cos(pi-(acos(1/7)-asin(6/14)))*8)))+atan((5+3+sin(pi-(acos(1/7)-asin(6/14)))*8)/(4*sqrt(10)-(cos(pi-(acos(1/7)-asin(6/14)))*8))))
Yields (not editing for proper formatting, paste into WolframAlpha):
tan^(-1)(1/8 (8 + 8 sin(cos^(-1)(1/7) - sin^(-1)(3/7))) sec(cos^(-1)(1/7) - sin^(-1)(3/7))) - tan^(-1)((8 + 8 sin(cos^(-1)(1/7) - sin^(-1)(3/7)))/(4 sqrt(10) + 8 cos(cos^(-1)(1/7) - sin^(-1)(3/7))))
=0.5680849462968405317515880584510907349252801422164102515326922606... radians.
