PhilonCubicEquation

Philon's problem

Given the point E, find the minimal in length secant FG, through E, intersecting the fixed lines OF, OG.

Here is a solution of the problem of Philon using functions and their graphs.
1) First show that the tan(fi), where fi is the angle of Philon's line with the x-axis, satisfies a cubic equation.
2) Taking the first line to be identical with the x-axis and the second be determined through a unit vector (c,s) the coefficients A, B, C, D of the equation depend on the location (x₀,y₀) of point E and on (c,s).
3) The dependence is given by the formulas, x, y replacing there the symbols x₀, y₀. f is an auxiliary quantity entering in the calculations.
4) The equation has always a unique root, represented here by x. x is the value of tan(fi) required.
5) With atan(x), the inverse tangens function, we find the slope of Philon's line.
6) Notice that EG = FJ, a fact proved in Philon.html .

Philon's problem

See Also