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.^[1][3]

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 ${\displaystyle DE}$ is the Philo line for point ${\displaystyle P}$ and angle ${\displaystyle DOE}$, and let ${\displaystyle Q}$ be the base of a perpendicular line ${\displaystyle OQ}$ to ${\displaystyle DE}$. Then ${\displaystyle DP=EQ}$ and ${\displaystyle DQ=EP}$.^[1]

Conversely, if ${\displaystyle P}$ and ${\displaystyle Q}$ are any two points equidistant from the ends of a line segment ${\displaystyle DE}$, and if ${\displaystyle O}$ is any point on the line through ${\displaystyle Q}$ that is perpendicular to ${\displaystyle DE}$, then ${\displaystyle DE}$ is the Philo line for angle ${\displaystyle DOE}$ and point ${\displaystyle P}$.^[1]

Doubling the cube[]

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 ${\displaystyle PQRS}$ be a rectangle whose aspect ratio ${\displaystyle PQ:QR}$ is ${\displaystyle 1:2}$, as in the figure. Let ${\displaystyle TU}$ be the Philo line of point ${\displaystyle P}$ with respect to right angle ${\displaystyle QRS}$. Define point ${\displaystyle V}$ to be the point of intersection of line ${\displaystyle TU}$ and of the circle through points ${\displaystyle PQRS}$. Because triangle ${\displaystyle RVP}$ is inscribed in the circle with ${\displaystyle RP}$ as diameter, it is a right triangle, and ${\displaystyle V}$ is the base of a perpendicular from the apex of the angle to the Philo line.

Let ${\displaystyle W}$ be the point where line ${\displaystyle QR}$ crosses a perpendicular line through ${\displaystyle V}$. Then the equalities of segments ${\displaystyle RS=PQ}$, ${\displaystyle RW=QU}$, and ${\displaystyle WU=RQ}$ follow from the characteristic property of the Philo line. The similarity of the right triangles ${\displaystyle PQU}$, ${\displaystyle RWV}$, and ${\displaystyle VWU}$ follow by perpendicular bisection of right triangles. Combining these equalities and similarities gives the equality of proportions ${\displaystyle RS:RW=PQ:QU=RW:WV=WV:WU=WV:RQ}$ or more concisely ${\displaystyle RS:RW=RW:WV=WV:RQ}$. Since the first and last terms of these three equal proportions are in the ratio ${\displaystyle 1:2}$, the proportions themselves must all be ${\displaystyle 1:{\sqrt[{3}]{2}}}$, 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]

References[]

Further reading[]

External links[]

• Weisstein, Eric W. "Philo Line". MathWorld.