documentclass[10pt]{amsart}
usepackage{mathtools,array}


usepackage{tikz}
usetikzlibrary{calc,positioning,intersections}




begin{document}

begin{tikzpicture}[x=0.25cm, y=0.25cm]

%The centers of a circle of radius 4 and a circle of radius 5 are at a distance of 16 from each other. Two line segments
%are to be drawn between them so that they are tangent to the circles. (It is a sketch of a conveyor belt.) The radii to
%the point of tangency of the same line segment are parallel to each other. The centers of the circles are located on the
%x-axis - the center of the circle of radius 5 is centered at the origin, and the center of the circle of radius 4 is
%centered at (16,0). If theta is the measure of the angle between the x-axis and the two radii, the slope of
%the line segment is sintheta/(16 - costheta). So, the slope of the radii is (costheta - 16)/sintheta. The slope is
%also tantheta.
%x = 5costheta.
%x^2 + ((costheta - 16)/sintheta)^2*x^2 = 5^2.
%tantheta=(costheta - 16)/sintheta.
%This is a quartic equation in the variable x. The solution is 5*sqrt(65/2 - 4*sqrt(66)). The point of tangency on the circle
%of radius 5 is
%(5*sqrt(65/2 - 4*sqrt(66)), 5*sqrt(4*sqrt(66) - 63/2)).
%
%
path (0,0) coordinate (center_of_first_circle) (16,0) coordinate (center_of_second_circle);
%
%
%
path let n1={5*sqrt(65/2 - 4*sqrt(66))}, n2={5*sqrt(4*sqrt(66) - 63/2)} in coordinate (a_point_of_tangency_on_bigger_circle) at (n1,n2);
path let n1={5*sqrt(65/2 - 4*sqrt(66))}, n2={5*sqrt(4*sqrt(66) - 63/2)} in coordinate (another_point_of_tangency_on_bigger_circle) at (n1,-n2);
%
path let n1={2*sqrt(66)}, n2={sqrt(64*sqrt(66)-504)} in coordinate (a_point_of_tangency_on_smaller_circle) at (n1,n2);
path let n1={2*sqrt(66)}, n2={sqrt(64*sqrt(66)-504)} in coordinate (another_point_of_tangency_on_smaller_circle) at (n1,-n2);
%
%
draw (a_point_of_tangency_on_bigger_circle) -- (a_point_of_tangency_on_smaller_circle);
draw (another_point_of_tangency_on_bigger_circle) -- (another_point_of_tangency_on_smaller_circle);
%
%
%
draw[blue] (center_of_first_circle) circle (5);
draw[blue]  (center_of_second_circle) circle (4);


end{tikzpicture}

end{document}