568 
PROBLEMS AND SOLUTIONS. 
[383 
but, observing that pqr = p x q 1 r 1 , we have 
(p-pOqin = _ (p -pùmpj 
P PPi 
hence 
c 2 p + b 2 v = — 
2 (p - Pi) (q ~ gi) (r - r^pspr, ' 
PPi 
and we have the like values for a?v + c 2 X and b-X + a?p respectively ; hence, omitting 
the symmetrical factor, we have, for the coordinates of the point in question, 
x : y : z — — : — : — 
PPi qqi rvi 
1_ 1 _1 
Taking the equation of the line Al to be Qy + Rz = 0, those of the lines II, IJ will be 
x = X (Qy + Rz), x = X' (Qy + Rz), 
where 
X Qß + Ry’ V Qß‘ + Ry ’ 
and the harmonic condition gives X + X' = 0, that is, 
Q (a/3' + a’fi) + R (or/ + a'y) = 0 ; 
the equation of the line Al is thus found to be 
(yci + y'a) y = (a/3 + a/3) z; 
and, since we have the like forms for the equations of the lines Bm and Cn, we have 
for the coordinates of the point of intersection of these three lines 
1 
1 
1 
x : y : z = 
ßy + ß'y yd + y'a ‘ aß' + a'ß ’ 
that is 
1 1 1 
p- + p/~ ’ q- + ql ’ r 2 + r/ 2 ' 
The equation of a conic circumscribed about the triangle ABO is 
where X, p, v are arbitrary coefficients; and the condition for the three points being 
in the conic is thus found to be 
(q - 2i) 2 > (r-ri) 2 =0, 
ppi , qqi , rr x 
p- + pi, q~ + ql, r 2 + rl