184 THE ROYAL SOCIETY OF CANADA 
Si 0 6 — 
bc(qr2— qri)sin A .e+ca(nipe—repi)sin B . e+ab(pig2 — prqi)sin C . es 
4A* 
5 (quro — Gori)er + (r1D2 — roi) € + (P1Q2 -- Pogi)es 
2A 
and e=xea+yetze in which x+y+s=], 
5 2Ax sin 0 = Qif2 — Gori 
2Ay sin 0=71p2—72p1 
2Az sin 0 = pige — po 
SO eee — A sree Cems | 
2A| Aan 2A,/A’,|h M M1 
pe qe M2 lo Mo Ne 



XII. Given the triangle ABC, A’ a point on BC, B’ a point CA, 
C’ a point on AB, G=CC’.AA’, E=AA’. BB’ and F=BB’ . CC’, 
to determine the ratios (A’B’C’) : (ABC) and (EFG) : (ABC). 
Let 
(AC’) :(C’B) ::m:X 
(BA’) : (A’C) ::m 2p 
(CB’) : (B’A) ::\:" 
ooh Pio AC ae BANC Bes C BAC BA 
and “AX:A+yty,::C’B.A’C:ac—C’B. BA’ 
N:Atutnm::C’B.CB’ :bc—B’A. AC’ 
and Aw: Ay wretyyy. : : A’C. CB’ :ab—A’'C. CB’ 
A'B'C’ _ (uet+nie) (Nate) (Aer t+ mez) 
ABC (un) (+n) (A+u)A 
27 (Au(r1-+ 72) €1€2€3 
(wt) Atm) Oma 
AC BAT. CB’+C'B .A'C.B'A 

abc 
EFG ce (Ne peo yes) (Ne mest ries) (Aria + urse + rires) 
ABC (A +u+ ve) (A+ u+v:) (Avy + uv + 172) A 
Au(ri — v.)e1ses 
2 (A+ u+re) (++) (ik uv rive) A 
