In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the Philo line is a
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
defined from an
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
and a
point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* 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
Philo of Byzantium ( el, , ''Phílōn ho Byzántios'', ca. 280 BC – ca. 220 BC), also known as Philo Mechanicus, was a Greek engineer, physicist and writer on mechanics, who lived during the latter half of the 3rd century BC. Although he was f ...
, a Greek writer on mechanical devices, who lived probably during the 1st or 2nd century BC. Philo used the line to
double the cube
Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related pro ...
; because doubling the cube cannot be done by a
straightedge and compass construction
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
, neither can constructing the Philo line.
Geometric characterization

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
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
to
. Then
and
.
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
.
Algebraic Construction
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
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
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
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore o ...
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
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
,
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
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
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
The Philo line can be used to
double the cube
Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related pro ...
, 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