106 On the Harmonic Relation of two Lines or two Points. 



gous relation between two lines with respect to a quadrilateral, 

 or between two points with respect to a quadrangle, i3 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 Traite des 

 Proprietes Projectives, and is, in fact, there substantially demon- 

 strated (see No. 163) ; and an explicit statement of the theo- 

 rem is given by M. Steiner, Lehrsatze 24 and 25, Crelle, vol. xiii. 

 p. 212 (a demonstration is given, vol. xix. p. 227). The theo- 

 rem 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 quadri- 

 lateral 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 dif- 

 ferent form. The three diagonals of the quadrilateral form a 

 triangle, the sides of which contain the six angles of the quadri- 

 lateral ; 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 are such that the lines joining them with the oppo- 

 site angles of the triangle meet in a point. Each of these points 

 is, with respect to the involution formed by the two angles of the 

 triangle, and the two points harmonically related thereto, a double 

 point ; and we have thus the following theorem of the harmonic 

 relation of two lines to a triangle and line, or else to a triangle 

 and point. 



Theorem. " If on the sides of a triangle there be taken three 

 points, which either lie in a line, or else are such that the lines 

 joining them with the opposite angles of a triangle meet in a 

 point ; and if on each side of the triangle there be taken two 

 points, forming with the two angles on the same side an involu- 

 tion having the first-mentioned point on the same side for a 

 double point ; then if three of the six points lie in a line, the 

 other three of the six points will also lie in a line," — the two lines 

 are said to be harmonically related to the triangle and line, or 

 (as the case may be) to the triangle and point. 



The theorems with respect to the harmonic relation of two 



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 be harmonically 

 related with respect to the triangle. 



