# Philo line

Philo's line can be used to double the cube, that is, to construct a geometric representation of the cube root of two, and this was Philo's purpose in defining this line (Coxeter and van de Craats, 1993). Specifically, let PQRS be a rectangle in which the aspect ratio PQ:QR is 1:2, as in the figure below. Let TU be the Philo line of point P with respect to right angle QRS. Define point V to be the point of intersection of line TU and of the circle through points PQRS, and let W be the point where line QR crosses a perpendicular line through V. Then segments RS and RW are in proportion ${\displaystyle \scriptstyle 1:{\sqrt[{3}]{2}}}$.