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# Philo Line

Given two intersecting lines and forming an angle with vertex at and a point inside the angle , the Philo line (or Philon line) is the shortest line segment touching both lines and passing through . 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 such that (Wells 1991).

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

where is the vertex angle and the point has polar coordinates .

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## References

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.

Philo Line

## Cite this as:

Weisstein, Eric W. "Philo Line." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PhiloLine.html