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 (x

3) The dependence is given by the formulas, x, y replacing there the symbols x

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 .

CubicSymmetry.html

Hyperbola.html

HyperbolaAsymptoticProperty.html

HyperbolaAsymptotics.html

HyperbolaProperty.html

HyperbolaPropertyMiddles.html

HyperbolaWRAsymptotics.html

Philon.html

PhilonCubic.html

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