THE ROYAL SOCIETY OF CANADA 
178 
PA OB MRC AE 
ca" TC ani BCE 
n — 
sin C 
* . by Lemma, Corollary, 
ues alm+bmm+cnms 
/{a(l—m) (l—n)+... } 
IV. Given the indices of the points A, B and C to determine the 
index of any line d in the plane of ABC. 
Let A”, B” and C” be the points of intersection of d and BC, CA 
and AB, and P, Q, and R the feet of perpendiculars on d from A, B 
and C respectively, let p= PA, g=QB and r=RC and letl:m, m:n, 
n:l be the Menelaän ratios of ABC : d, then will 
(CYA) _(C’P) _ (PA) 
l 
(C'B”) (C’Q) (QB) m 
(A'B)_(A'Q) (QB) _ m 
(AC) (A’R) (RC) x 
(B”C) __ (BR) _ (RO) _ 
(B'A) (B’P) (PA) ; 
(A”B”) = (A’R) 4 (RB”) 
(RC) | RC) RO 
_(4"Q), (PB") 
~ (QB) (PA) 




AVERT As PB" 
AB! _4"Q | PE" 
r q ? 
: / 
Let &, €2, €3, €'1, €'2, es, m, 12, 73 and n be the indices of A, B, C, 
A”, B”, C’,.BC, CA, AB and d respectively then will 
ea bes q€3 — Ve 
g—7 
oe re, — Per 
ey 
. _ 1 (peres+geser trees) 
(g—r) (r—P) 
Peres t+ Qesei + rec 
A"B"n — \ 
b+q—r—pgr-* 
