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Wikipedia / 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; because doubling the cube cannot be done by a straightedge and compass construction, neither can constructing the Philo line.

Geometric characterization

be the base of a perpendicular line.

.

Algebraic Construction

in that infinite triangle is obtained by the following algebra:

is the tangent on the other side of the triangle.

is

These lines intersect the horizontal axis at

which has the solution

at

which has the solution

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

.

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.]

.

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

Q

yields

is

  1. .

is

  1. .

The difference of these two expressions is

  1. .

.

Special case: right angle

of the previous section results in the following special case:

-axis at

which has the solution

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

, so

equivalent to

Therefore

and plugging this into another equation above one finds

Doubling the cube

The Philo line can be used to double the cube, that is, to construct a geometric representation of the cube root be a rectangle whose aspect ratio is the base of a perpendicular from the apex of the angle to the Philo line.

, the proportion that is required to double the cube.

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

Minimizing the area

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

  1. .

,

  1. .

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

and the minimum area

  1. .

References

Further reading