The Transversal "AB" drawn through a given point
"P" within a given angle "RCS" so that the sides of the angle intercept
on the transversal a segment of minimum length has become known as the
Philo Line (or Philon Line) of the point for the given angle. The line
is name after Philon of Byzantium,
an ancient Greek engineer. This problem was devised by Philo as a
reduction to the duplication
of the cube problem, and because of this link to this famous problem,
it has excited interest over the ages.
References
[1] Howard Eves, Scripta Mathematica, vol.26, 1959, pp. 141-148
[2] Sir Thomas L. Heath, A Manual of Greek Mathematics, Oxford University Press, 1963, pp.262-264
[3] B. Bhattacharya and G. Toussaint, Computing Shortest Transversals, Computing, vol. 46, 1991, pp 93-119
[4] Coxeter and Van de Craats, Philon Lines In Non-Euclidean Planes, Journal of Geometry, vol. 48, 1993, pp 26-55
[5] David Hounshell, Grolier Electronic Publishing, 1995.