Philo line

In geometry, the Philo line is a line segment defined from an angle and a point. The Philo line for a point "P" that lies inside an angle with edges "d" and "e" is the shortest line segment that passes through "P" and has its endpoints on "d" and "e". Also known as the Philon line, it is named after Philo of Byzantium, a Greek writer on mechanical devices, who lived probably during the first or second century BC. The Philo line is not, in general, constructible by compass and straightedge.

Doubling the cube

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 $scriptstyle\; 1:sqrt\; [3]\; \{2\}$.

In this figure, segments "PU" and "VT" are of equal length, and "RV" is perpendicular to "TU". These properties can be used as part of an equivalent alternative definition for the Philo line for a point "P" and angle edges "d" and "e": it is a line segment connecting "d" to "e" through "P" such that the distance along the segment from "P" to "d" is equal to the distance along the segment from "V" to "e", where "V" is the closest point on the segment to the corner point of the angle.

Since doubling the cube is impossible with compass and straightedge, it is similarly impossible to construct the Philo line with these tools.

References

*cite journal
author = Coxeter, H. S. M.; van de Craats, Jan
title = Philon lines in non-Euclidean planes
journal = Journal of Geometry
volume = 48
year = 1993
issue = 1–2
pages = 26–55
id = MathSciNet | id = 1242701
doi = 10.1007/BF01226799

*cite journal
author = Eves, Howard
authorlink = Howard Eves
title = Philo's line
journal = Scripta Mathematica
volume = 24
year = 1959
pages = 141–148
id = MathSciNet | id = 0108755

*cite book
author = Eves, Howard
authorlink = Howard Eves
title = A Survey of Geometry
edition = vol. 2
publisher = Allyn and Bacon
location = Boston
year = 1965
pages = 39, 234–236

*cite book
author = Kimberling, Clark
title = Geometry in Action: A Discovery Approach Using The Geometer's Sketchpad
publisher = [http://www.keycollege.com/ Key College Publishing]
location = Emeryville, California
pages = 115–6
id = ISBN 1-931914-02-8
year = 2003

*cite journal
author = Neovius, Eduard
title = Ueber eine specielle geometrische Aufgabe des Minimums
journal = Mathematische Annalen
volume = 31
year = 1888
pages = 359–362
doi = 10.1007/BF01206220

*cite journal
author = Neuberg, J.
title = Sur un minimum
journal = Mathesis
year = 1907
pages = 68–69

*cite journal
author = Wetterling, W. W. E.
title = Philon's line generalized: an optimization problem from geometry
journal = Journal of Optimization Theory and Applications
volume = 90
year = 1996
issue = 3
pages = 517–521
id = MathSciNet | id = 1402620
doi = 10.1007/BF02189793

External links

*mathworld | title = Philo Line | urlname = PhiloLine