[size=150]In geometry, the Philo line is a line segment defined from an angle and a point inside the angle as the shortest line segment through the point that has its endpoints on the two sides of the angle. Also known as the Philon line, it is named after Philo of Byzantium, a Greek writer on mechanical devices, who lived probably during the 1st or 2nd century BC. Philo used the line to double the cube; because doubling the cube cannot be done by a straightedge and compass construction, neither can constructing the Philo [br]line.[br][br]Geometric characterization. The Philo line of a point P and angle DOE, and the defining equality of distances from P and Q to the ends of DE, where Q is the base of a perpendicular from the apex of the angle. The defining point of a Philo line, and the base of a perpendicular from the apex of the angle to the line, are equidistant from the endpoints of the line. That is, suppose that segment is the Philo line for point {\displaystyle P}[img]https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a[/img] and angle, and let Q be the base of a perpendicular line. Then [img]https://wikimedia.org/api/rest_v1/media/math/render/svg/77491d9c4f9b9015de754dc28090aa3256fc8355[/img] and DQ=EP.Conversely, if P and Q are any two points equidistant from the ends of a line segment, and if {Q is any point on the line through Q that is perpendicular to [img]https://wikimedia.org/api/rest_v1/media/math/render/svg/3c123d37c1276b5d6df0a6328c59167fc5ed82bd[/img], then [img]https://wikimedia.org/api/rest_v1/media/math/render/svg/3c123d37c1276b5d6df0a6328c59167fc5ed82bd[/img] is the Philo line for angle DOE and point [img]https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a[/img].[br][/size]