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| In [[geometry]], the '''Philo line''' is a [[line segment]] defined from an [[angle]] and a [[point (geometry)|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 1st or 2nd century BC. The Philo line is not, in general, [[Constructible number|constructible]] by [[compass and straightedge]].
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| == Doubling the cube ==
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| Philo's line can be used to [[doubling the cube|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 <math>\scriptstyle 1:\sqrt[3]{2}</math>.
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| <center>[[Image:Philo line.svg|360px]]</center>
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| 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.
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| Since doubling the cube is impossible with [[compass and straightedge]], it is similarly impossible to construct the Philo line with these tools.
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| == References ==
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| <div class="references-small" style="-moz-column-count:2; column-count:2;">
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| *{{cite journal
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| | author = [[H. S. M. Coxeter|Coxeter, H. S. M.]]; van de Craats, Jan
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| | title = Philon lines in non-Euclidean planes
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| | journal = [[Journal of Geometry]]
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| | volume = 48
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| | year = 1993
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| | issue = 1–2
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| | pages = 26–55
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| | id = {{MathSciNet | id = 1242701}}
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| | doi = 10.1007/BF01226799}}
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| *{{cite journal
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| | author = Eves, Howard
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| | authorlink = Howard Eves
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| | title = Philo's line
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| | journal = [[Scripta Mathematica]]
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| | volume = 24
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| | year = 1959
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| | pages = 141–148
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| | id = {{MathSciNet | id = 0108755}}}}
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| *{{cite book
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| | author = Eves, Howard
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| | authorlink = Howard Eves
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| | title = A Survey of Geometry
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| | edition = vol. 2
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| | publisher = Allyn and Bacon
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| | location = Boston
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| | year = 1965
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| | pages = 39, 234–236}}
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| *{{cite book
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| | author = Kimberling, Clark
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| | title = Geometry in Action: A Discovery Approach Using The Geometer's Sketchpad
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| | publisher = [http://www.keycollege.com/ Key College Publishing]
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| | location = Emeryville, California
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| | pages = 115–6
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| | isbn = 1-931914-02-8
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| | year = 2003}}
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| *{{cite journal
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| | author = Neovius, Eduard
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| | title = Ueber eine specielle geometrische Aufgabe des Minimums
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| | journal = [[Mathematische Annalen]]
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| | volume = 31
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| | year = 1888
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| | pages = 359–362
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| | doi = 10.1007/BF01206220
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| | issue = 3}}
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| *{{cite journal
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| | last = Neuberg | first = J. | authorlink = Joseph Jean Baptiste Neuberg
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| | title = Sur un minimum
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| | journal = [[Mathesis (journal)|Mathesis]]
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| | year = 1907
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| | pages = 68–69}}
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| *{{cite journal
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| | author = Wetterling, W. W. E.
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| | title = Philon's line generalized: an optimization problem from geometry
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| | journal = [[Journal of Optimization Theory and Applications]]
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| | volume = 90
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| | year = 1996
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| | issue = 3
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| | pages = 517–521
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| | id = {{MathSciNet | id = 1402620}}
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| | doi = 10.1007/BF02189793}}
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| </div>
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| == External links ==
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| *{{mathworld | title = Philo Line | urlname = PhiloLine}}
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| {{DEFAULTSORT:Philo Line}}
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| [[Category:Euclidean plane geometry]]
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Nice to meet you, my name is Figures Held though I don't truly like becoming known as like that. To gather badges is what her family members and her appreciate. Supervising is my profession. Her family members lives in Minnesota.
Feel free to surf to my blog post www.ddhelicam.com