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Brachistochrone curve from the origin to the point A

Shown is the Brachistochrone curve that starts from the origin and goes through the movable point A. Try clicking on and dragging the point A to see how the shape of the curve changes. Note that C/2 is the radius of the rolling circle whose cycloid forms the Brachistochrone (see "Calculus of Variations" p38-39, Lev D. Elsgolc).
Steps to create Set up a moveable point A. Input line: d:=y(A) / x(A) e:=Element(NSolve(cos(x) - 1 = d (x - sin(x))), 1) Create a point B as the Intersection of the xAxis and the line e, and hide them both. Input line: C:=2y(A) / (cos(x(B)) - 1) f(t):=Curve((C / 2 (t - sin(t)), C / -2 (1 - cos(t))), t, 0, 2π)