Taking the parallelogram a rectangle, or the triangle isosceles, we get:
Any two lines and the bisections of their angles form an harmonic pencil. Or:
In an harmonic pencil, if two conjugate rays are perpendicular, then the other two are equally inclined to them; and, conversely, if one ray bisects the angle between conjugate rays, it is perpendicular to its conjugate.
This connects perpendicularity and bisection of angles with projective properties.
§ 24. We add a few theorems and problems which are easily proved or solved by aid of harmonics.
An harmonic pencil is cut by a line parallel to one of its rays in three equidistant points.
Through a given point to draw a line such that the segment determined on it by a given angle is bisected at that point.
Having given two parallel lines, to bisect on either any given segment without using a pair of compasses.
Having given in a line a segment and its middle point, to draw through any given point in the plane a line parallel to the given line.
To draw a line which joins a given point to the intersection of two given lines which meet off the drawing paper (by aid of § 21).