96
[118
118.
ON THE HARMONIC RELATION OF TWO LINES OR TWO
POINTS.
[From the Philosophical Magazine, vol. vi. (1853), pp. 105—107.]
The “ harmonic relation of a point and line with respect to a triangle ” is well
known and understood 1 ; but the analogous relation between two lines with respect to
a quadrilateral, or between two points with respect to a quadrangle, is not, I think,
sufficiently singled out from the mass of geometrical theorems so as to be recognized
when implicitly occurring in the course of an investigation. The relation in question,
or some particular case of it, is of frequent occurrence in the Traité des Propriétés
Projectives, [Paris, 1822], and is, in fact, there substantially demonstrated (see No. 163) ;
and an explicit statement of the theorem is given by M. Steiner, Lehrsätze 24 and 25,
Grelle, t. XIII. [1835] p. 212 (a demonstration is given, t. xix. [1839] p. 227). The
theorem containing the relation in question may be thus stated.
Theorem of the harmonic relation of two lines with respect to a quadrilateral. “ If
on each of the three diagonals of a quadrilateral there be taken two points harmonically
related with respect to the angles upon this diagonal, then if three of the points lie
in a line, the other three points will also lie in a line”—the two lines are said to
be harmonically related with respect to the quadrilateral.
It may be as well to exhibit this relation in a somewhat different form. The
three diagonals of the quadrilateral form a triangle, the sides of which contain the
six angles of the quadrilateral ; and considering three only of these six angles (one
angle on each side), these three angles are points which either lie in a line, or else
1 The relation to which I refer is contained in the theorem, “If on each side of a triangle there be
taken two points harmonically related with respect to the angles on this side, then if three of these points
lie in a line, the lines joining the other three points with the opposite angles of the triangle meet in a
■point,”—the line and point are said to he harmonically related with respect to the triangle.
