[GLASHAN] ALTERNATE NUMBERS AS GEOMETRICAL INDICES 177 
III. Given the indices of the lines AB, BC and CA, to determine 
the index of any other line d, in the plane of ABC.’ 
Let d intersect BC in A”, CA in B” and AB in C’”, let P, QandR 
be the feet of perpendiculars let fall on d from A, B and C respectively 
and let 1:m, m:n and n :l be the Menelaän ratios of (ABC; d), 
then will 
(C’A) (PA) _ / 
(C"B) (QB) m 
(A’C) (RC) un 
(BYC) (RC) x 
(B’A) (PA) 1! 
Draw CC” and let P’ and Q’ 
be the feet of perpendiculars let 
fall on CC” from A and B, and 
m, n2, 73, n’ and 7 be the in- 
dices of BC, CA, AB, CC” and 
A’C” respectively, then will 
a sin(CC” ZCA)m-+sin(BC Z CC") ne 
i sin(BC Z CA) 
sin C’CAm+sin C’CBm 
sin C 

sin(A"C" Z AB)n'+sin(CC" ZA"C") ns 
sin(CC” Z AB) 
sin(A”C” ZAB)n’—sin A”C"Cn3 
sin(CC” Z AB) 
PAS PA OB OB RE 
ET, Lust? SU? ae 
CA ca ie Be" co" 
sin(CC” ZAB)sin C 
and nn — 
Also, sin(CC” ZAB) =—— = =— = 
