30 
ON SOME GEOMETRICAL THEOREMS RELATING TO 
[165 
The harmonic of the point 0 with respect to the triangle is a line (o), which 
is also the polar of 0 with respect to the conic. The conic meets this line in two- 
points (as the figure is drawn, imaginary ones), I, J. 
Suppose now that the harmonic lines (r), (s) are harmonically related to the points 
I, J, (i.e. let the lines (r), (s) and the lines through their point of intersection and 
I, J, be a harmonic pencil, (or, what is the same thing, let I, J, and the points in 
which the line of junction meets the lines (r), (s) be a harmonic range), then,. 
Theorem, the point of intersection of the lines (?’), (s) lies on the conic. 
In order to prove this, take 
x = 0, y — 0, z = 0, for the equations of BG, CA, AB. 
x = y = z, for the point 0, 
the equations of the harmonic lines will be 
ax + hy f cz — 0, 
x 
a 
y 2 K 
+ f + - = 0, 
b c 
the equation of the conic is 
x 2 + y 1 + z 1 — 2 yz — 2 zx — 2 xy = 0 ; 
the equation of the line (o) is 
x + y + z = 0. 
The coordinates of the points /, J, are 
and 
x : y : z = 1 : œ : <o 2 , 
x : y : z = 1 : co 2 : co, 
where to is an imaginary cube root of unity. 
The equations of the lines joining the point of intersection of (r), (s) with the 
points I, J, are 
ax + by + cz _ bcx + cay + abz 
a + bco + cco be + caw + abw 2 ’ 
and 
ax+by +cz _ bcx + cay + abz 
a + bor + cw be + caw 2 + abw ’ 
and these will form with the lines 
ax + by + cz = 0, 
bcx + cay + abz = 0,