118] ON THE HARMONIC RELATION OF TWO LINES OR TWO POINTS. 97 
are such that the lines joining them with the opposite 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 involution 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 points are of course 
the reciprocals of those with respect to the harmonic relation of two lines, and do 
not need to be separately stated. 
The preceding theorems are useful in (among other geometrical investigations) the 
porism of the in-and-circumscribed polygon. 
2 Stone Buildings, July 9, 1853. 
C. II. 
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