383] PROBLEMS AND SOLUTIONS, 
569 
but, in virtue of the relations 
(p-Pif = - %ppi + (p 2 +Pi% &c., 
this equation is identically true, and the subsidiary theorem is thus proved. 
Passing now to the general theorem, I prove the first part of it as follows: 
The equation of a conic circumscribed about the triangle x = 0, y = 0, z = 0 is 
the conic, we have 
and thence 
ABC 
H : 
a P 7 
A 
B 
(7 
— 
3 h 
— = 
X 
y 
z 
/3", 
7") are 
A 
B 
C _ 
-f 'ft, H" 
/ ” 
a! 
/3 
7 
1 
1 
1 
a 
’ ¡9 ’ 
7 
1 
1 
1 
a' 
’ w 
7 
1 
1 
1 
a" 
’ /8"’ 
// 
7 
A B C_ 
a" W 
= 0, 
which is the condition for the intersection in a point of the three lines 
x y z _ 
- + | + - = °> 
« Æ 7 
# y z 
> + J57 H 7 — 0> 
« P 7 
X y z _ 
1- — -1 = 0 • 
a /3" 7" ’ 
and the theorem in question is thus proved. I remark, in passing, that the theorem 
might also be stated as follows :—The locus of a point 0, such that its polar in 
regard to the triangle ABC passes through a fixed point il, is a conic circumscribed 
about the triangle. 
To prove the second part of the theorem, take for the coordinates of the points 
0, O', 0" respectively (a, /3, 7), (a', /3', y), (a", ¡3", 7") ; then 
= 0, 
c. V. 
72