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.
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 .
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 .
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 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.
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