ON SOME GEOMETRICAL THEOREMS RELATING TO A TRIANGLE 
CIRCUMSCRIBED ABOUT A CONIC. 
[From the Quarterly Mathematical Journal, vol. i. (1857), pp. 169—175.] 
The following investigations were suggested to me by Sir F. Pollock’s interesting 
paper “ On a Geometrical Theorem relating to an Equilateral Triangle circumscribed 
about a Circle,” [Quart. Math. Jour. t. I. (1857), pp. 167—169]. 
If on the sides of a triangle ABC, there be taken points a, /3, 7, such that 
Act, B/3, C>y meet in a point 0; 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 two lines are said to be harmonically related with respect to 
the triangle ABC and point 0. Call these the lines (r), (s). 
The triangle ABC and point 0 give rise to a determinate conic; viz., the conic 
touching the sides at the points a, /3, 7.