Tools: Angle Chords

John McCoy

Reading Apollonius' Conics is a daunting task. Geometry of his day was a tool for thinking and even though his first books primarily document the then state of the art relative to conics it was not written for novice geometers. Rather, it was addressed to journeyman geometers who could benefit from studying it. Few today fall into that group.

This document is an attempt to provide enough background to enable the reader to develop sufficient understanding of concepts and ideas from the first book that they become a useful part of the readers "thinking repertoire."  Particularly problem-some are propositions and methods that Apollonius and others used without comment.  They were assumed to be known and no doubt they were at the time a part of any journeyman's knowledge bank.  To the uninitiated though they constitute black magic.

We begin with one such proposition that I have only found in Heath's introduction to the conics.

"if two chords drawn in fixed directions between two lines forming an angle intersect in a point the ratio of the rectangles under the segments is independent of the position of the point."

A proposition dealing with segments of lines intersected by parallel lines exists and is attributed to Thales but I can find no such attribution regarding this theorem. We choose therefore to refer to this simply as the "intersecting chords of an angle" theorem.

The figure to the left is from a GeoGebra model. $\overline{FE}$ and $\overline{BC}$ are chords between the two lines that form an angle at point $A$. They are chords drawn in fixed directions and intersect at point $L$. The theorem states that the lines $\overline{FE}$ and $\overline{BC}$ can be moved and their point of intersection changed and, so long as the directions of the lines are not changed, the ratio $\frac{rect(\overline{BL},\overline{LC})} {rect(\overline{FL},\overline{LE})}$ will not change.

The two parts of the figure differ only in that the two lines $\overline{FE}$ and $\overline{BC}$ have been moved without changing their directions and show that the ratio $\frac{rect(\overline{BL},\overline{LC})}{rect(\overline{FL},\overline{LE})}$is not changed by the move.

$\frac{rect(\overline{BL},\overline{LC})} {rect(\overline{FL},\overline{LE})}$ is a literal interpretation of the method used to make the comparison. Rectangles were constructed and their areas used to divide a line proportionally as the areas of the rectangles.

In the next figure the model has been extended to segments of external intersections of the chords. Lines $\overline{BC}$and $\overline{FE}$ can be moved by dragging points $F$ and $C$ and their directions can by changed by dragging the small red and blue circular handles attached to points $C$ and $F$. A line parallel to $\overline{FE}$ has been drawn though point $A$ and intersects the extension of $\overline{BC}$ at $K$.  The model can be opened in your browser using this link: Angle Chord Intersections

If point $F$ is used to drag the line $\overline{FE}$ up to point $A$ where $\overline{FL} = \overline{EL} = \overline{AK}$ it leads immediately to $\frac{\overline{EL}*\overline{FL}}{\overline{CL}*\overline{BL}}=\frac{\overline{AK}^2}{\overline{BK}*\overline{CK}}$.

Apollonius uses this relationship in his proof that sections cut in a certain way from a cone are ellipse. Here we only show that the observation that the ratio is a constant can be proven geometrically. Paraphrasing Archimedes, "It's easier to prove something when you have discovered mechanically how it works."

$\overline{AK} \parallel \overline{FL} $ and $\triangle ECL$ is similar to $\triangle ACK$ therefore $\frac{\overline{EL}}{\overline{AK}}=\frac{ \overline{CL}}{\overline{CK}}.$  $\triangle FBL$ is similar to  $\triangle ABC$ therefore $\frac{\overline{FL}}{\overline{AK} }=\frac{ \overline{BL}}{\overline{BK}}.$

Multipling the two equalities $\frac{\overline{EL}*\overline{FL}}{\overline{AK}*\overline{AK}}=\frac{\overline{CL}*\overline{BL}}{\overline{BK}*\overline{CK}}$ the product can be rearranged  to $\frac{\overline{EL}*\overline{FL}}{\overline{CL}*\overline{BL}}=\frac{\overline{AK}^2}{\overline{BK}*\overline{CK}}$ as discovered above.

There are two chords that bear particular interest. They are the pair of chords that make equal but opposite angles with the angle bisector (central axis) as shown in this figure. They also make equal angles with the perpendicular to the central axis and are refered to as anti-parallels.  These particular chords are of interest because the ratios of the products of their segments is 1.

Equally interesting is that although the pair of chords may intersect either internally or externally to the sides of the angle the mid point of the chords lie on lines from the angle apex that form equal angles with but on opposite sides of the central axis.  The two chords comprising the pair are members of families of parallel chords centered on these lines.

When the intersections of a chord and the sides of the angle are on opposite sides of the angle's vertex, as shown at the left, the ratio of the products of the chord segments is still independent of the position of the intersection and that constant value remains $\frac{\overline{EL}*\overline{FL}}{\overline{CL}*\overline{BL}}=\frac{\overline{AK}^2}{\overline{BK}*\overline{CK}}$

Where $\overline{AK}$ is drawn parallel to $\overline{ELF}$.

Apollonius uses this version of the theorem in his proof that certain sections cut from a cone are hyperbola.

jhmc 2014-10-30