60 
ON A LOCUS IN RELATION TO THE TRIANGLE. 
[394 
contains the angles of the reciprocal triangle, and is thus in fact the conic in which 
are situate the angles of the two triangles. For the coordinates of one of the angles 
of the reciprocal triangle are (A, H, G); we should therefore have 
~HG + ~GA +%AH = 0, 
/ 9 h 
which is 
j^iGHgh + BGhf + GHfg) = 0, 
or attending only to the second factor and writing 
GH = Kf+AF, 
the condition is 
Kfgh 4- AFgh + BGhf+ CHfg = 0, 
or substituting for K, A, B, C, F, G, H their values and reducing, this is 
7 ,, 0 ,, / 1 1 1 1 2\ A 
- U - - a p - hg , - cT , +_pj = 0, 
which is satisfied: hence the three angles of the reciprocal triangle lie on the conic 
in question. 
Partially recapitulating the foregoing results, we see in the case where the Absolute 
is not a point-pair, that the locus of a point such that the perpendiculars from it 
on the sides of the triangle have their feet in a line, is in general a cubic curve 
passing through the angles of the triangle : if, however, the condition 
1 _ 1 _ 1 _ 1 2_ 
abc a/ 2 bg 2 ch 2 **" fgh 
be satisfied, that is, if the triangle be such that the angles thereof and of the 
reciprocal triangle lie in a conic (or, what is the same thing, if the sides touch a 
conic) then the cubic locus breaks up into the line % + - + = 0, which is the line 
/ 9 h 
through the points of intersection of the corresponding sides of the two triangles, and 
into the conic 
A B C 
jyz+-zx + ^ay=Q, 
which is the conic through the angles of the two triangles. 
The question arises, given a conic (the Absolute) to construct a triangle such that 
its angles, and the angles of the reciprocal triangle in regard to the given conic, lie 
in a conic.