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;[1][2] because doubling the cube cannot be done by a straightedge and compass construction, neither can constructing the Philo line.[1][3]

Geometric characterization

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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   and angle  , and let   be the base of a perpendicular line   to  . Then   and  .[1]

Conversely, if   and   are any two points equidistant from the ends of a line segment  , and if   is any point on the line through   that is perpendicular to  , then   is the Philo line for angle   and point  .[1]

Algebraic Construction

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A suitable fixation of the line given the directions from   to   and from   to   and the location of   in that infinite triangle is obtained by the following algebra:

The point   is put into the center of the coordinate system, the direction from   to   defines the horizontal  -coordinate, and the direction from   to   defines the line with the equation   in the rectilinear coordinate system.   is the tangent of the angle in the triangle  . Then   has the Cartesian Coordinates   and the task is to find   on the horizontal axis and   on the other side of the triangle.

The equation of a bundle of lines with inclinations   that run through the point   is

 

These lines intersect the horizontal axis at

 

which has the solution

 

These lines intersect the opposite side   at

 

which has the solution

 

The squared Euclidean distance between the intersections of the horizontal line and the diagonal is

 

The Philo Line is defined by the minimum of that distance at negative  .

An arithmetic expression for the location of the minimum is obtained by setting the derivative  , so

 

So calculating the root of the polynomial in the numerator,

 

determines the slope of the particular line in the line bundle which has the shortest length. [The global minimum at inclination   from the root of the other factor is not of interest; it does not define a triangle but means that the horizontal line, the diagonal and the line of the bundle all intersect at  .]

  is the tangent of the angle  .

Inverting the equation above as   and plugging this into the previous equation one finds that   is a root of the cubic polynomial

 

So solving that cubic equation finds the intersection of the Philo line on the horizontal axis. Plugging in the same expression into the expression for the squared distance gives

 

Location of  

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Since the line   is orthogonal to  , its slope is  , so the points on that line are  . The coordinates of the point   are calculated by intersecting this line with the Philo line,  .   yields

 
 

With the coordinates   shown above, the squared distance from   to   is

 .

The squared distance from   to   is

 .

The difference of these two expressions is

 .

Given the cubic equation for   above, which is one of the two cubic polynomials in the numerator, this is zero. This is the algebraic proof that the minimization of   leads to  .

Special case: right angle

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The equation of a bundle of lines with inclination   that run through the point  ,  , has an intersection with the  -axis given above. If   form a right angle, the limit   of the previous section results in the following special case:

These lines intersect the  -axis at

 

which has the solution

 

The squared Euclidean distance between the intersections of the horizontal line and vertical lines is

 

The Philo Line is defined by the minimum of that curve (at negative  ). An arithmetic expression for the location of the minimum is where the derivative  , so

 

equivalent to

 

Therefore

 

Alternatively, inverting the previous equations as   and plugging this into another equation above one finds

 

Doubling the cube

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The Philo 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. Specifically, let   be a rectangle whose aspect ratio   is  , as in the figure. Let   be the Philo line of point   with respect to right angle  . Define point   to be the point of intersection of line   and of the circle through points  . Because triangle   is inscribed in the circle with   as diameter, it is a right triangle, and   is the base of a perpendicular from the apex of the angle to the Philo line.

Let   be the point where line   crosses a perpendicular line through  . Then the equalities of segments  ,  , and   follow from the characteristic property of the Philo line. The similarity of the right triangles  ,  , and   follow by perpendicular bisection of right triangles. Combining these equalities and similarities gives the equality of proportions   or more concisely  . Since the first and last terms of these three equal proportions are in the ratio  , the proportions themselves must all be  , the proportion that is required to double the cube.[4]

 

Since doubling the cube is impossible with a straightedge and compass construction, it is similarly impossible to construct the Philo line with these tools.[1][3]

Minimizing the area

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Given the point   and the angle  , a variant of the problem may minimize the area of the triangle  . With the expressions for   and   given above, the area is half the product of height and base length,

 .

Finding the slope   that minimizes the area means to set  ,

 .

Again discarding the root   which does not define a triangle, the slope is in that case

 

and the minimum area

 .

References

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  1. ^ a b c d e Eves, Howard (1965). A Survey of Geometry. Vol. 2. Boston: Allyn and Bacon. pp. 39, 234–236.
  2. ^ Wells, David (1991). "Philo's line". The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books. pp. 182–183.
  3. ^ a b Kimberling, Clark (2003). Geometry in Action: A Discovery Approach Using The Geometer's Sketchpad. Emeryville, California: Key College Publishing. pp. 115–116. ISBN 1-931914-02-8.
  4. ^ Coxeter, H. S. M.; van de Craats, Jan (1993). "Philon lines in non-Euclidean planes". Journal of Geometry. 48 (1–2): 26–55. doi:10.1007/BF01226799. MR 1242701. S2CID 120488240.

Further reading

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