Philo Line


Given two intersecting lines OA and OB forming an angle with vertex at O and a point X inside the angle ∠AOB, the Philo line (or Philon line) is the shortest line segment AB touching both lines and passing through X. The line is named for Philo of Byzantium who considered the line while attempting to duplicate the cube. The line can be constructed by finding OY_|_AB such that AX=BY (Wells 1991).


The distances along the angle edges x and h and the lengths along the Philo line l and dl can be computed by solving the simultaneous equations


where theta is the vertex angle and the point X has polar coordinates (r,phi).

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Eves, H. "Philo's Line." Scripta Math. 24, 141-148, 1959.Eves, H. W. A Survey of Geometry, Vol. 2. Boston, MA: Allyn and Bacon, pp. 39 and 234-238, 1965.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 182-183, 1991.Wells, D. G. You Are a Mathematician: A Wise and Witty Introduction to the Joy of Numbers. New York: Wiley, 1997.

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Philo Line

Cite this as:

Weisstein, Eric W. "Philo Line." From MathWorld--A Wolfram Web Resource.

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