Philo line

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.

are any two points equidistant from the ends of a line segment

[1] A suitable fixation of the line given the directions from

in that infinite triangle is obtained by the following algebra: The point

is put into the center of the coordinate system, the direction from

The equation of a bundle of lines with inclinations

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.

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

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.

is The difference of these two expressions is Given the cubic equation for

The equation of a bundle of lines with inclination

of the previous section results in the following special case: These lines intersect the

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 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.

is the base of a perpendicular from the apex of the angle to the Philo line.

follow from the characteristic property of the Philo line.

follow by perpendicular bisection of right triangles.

Since the first and last terms of these three equal proportions are in the ratio

, 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.

, a variant of the problem may minimize the area of the triangle

given above, the area is half the product of height and base length, Finding the slope

that minimizes the area means to set

which does not define a triangle, the slope is in that case and the minimum area