3.94] 
53 
394. 
ON A LOCUS IN RELATION TO THE TRIANGLE. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. vm. (1867), 
pp. 261—277.] 
If from any point of a circle circumscribed about a triangle perpendiculars are 
let fall upon the sides, the feet of the perpendiculars lie in a line; or, what is the 
same thing, the locus of a point, such that the perpendiculars let fall therefrom upon 
the sides of a given triangle have their feet in a line, is the circle circumscribed 
about the triangle. 
In this well known theorem we may of course replace the circular points at 
infinity by any two points whatever; or the -Absolute being a point-pair, and the 
terms perpendicular and circle being understood accordingly, we have the more general 
theorem expressed in the same words. 
But it is less easy to see what the corresponding theorem is, when instead of 
being a point-pair, the Absolute is a proper conic; and the discussion of the question 
affords some interesting results. 
Take (cc = 0, y = 0, 2 = 0) for the equations of the sides of the triangle, and let 
the equation of the Absolute be 
(a, b, c, f g, K$x, y, zf = 0, 
then any two lines which are harmonics in regard to this conic (or, what is the 
same thing, which are such that the one of them passes through the pole of the 
other) are said to be perpendicular to each other, and the question is: 
Find the locus of a point, such that the perpendiculars let fall therefrom on the 
sides of the triangle have their feet in a line. 
Supposing, as usual, that the inverse coefficients are (A, B, C, F, G, H), and that 
K is the discriminant, the coordinates of the poles of the three sides respectively are