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 
.
