[GLAsHAN] ALTERNATE NUMBERS AS GEOMETRICAL INDICES 185 
4 (AC. BA’ .CB’=C'B.A'C .B’'A)* 
(ab—A’C. CB’) (bc—B'A . AC’) (ca—C’B .. BA’) 
These ratios were given without demonstrations, in Routh’s 
Analytical Statics, Vol. I, p. 89, edition of 1891. 
XIII. Given the coplanar triangles ABC and EFG and the points 
A’=BC.AE, B’=CA. BF and C’=AB. CG, to determine the ratio 
(EFG) : (ABC). 
Let 
(AC’) :(C’B) ::u:d 
(BA’) : (A’C) :: m:n 
(CB’) : (B’A) ::\:» 
(AE) :(EBA’) ::u+n:p - 
(BF) : (FB’) ::A\+n:6 
(CG) :(GC’) : :A+u:7r 
then will 
à pa + Mes v163 
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EE Xe + cet ve 
À o + 2 
Ce a + cent Te; 
Aer 
[Det Lal 
AN CF M» 
Ney 
os WIIG) AC051 Oe re ee 
(eut) (Ato+r) (A+u+7) 
BC LAS AC «An BAY Ar 
CB’ OBE CA FB BAL BF 
Bye CG. ACO CG ABs GC 
Albee Contin AA BB CC! 
This theorem includes both (i) and (ii) of Prop. XII; 
if HA’ = FB'=GC'=0, it reduces to (i); 
AB x BC BLA Bie CA, CB d CG AB ALG 
PA 2 CBB PBS GR AAC GC = AC BA! 
it reduces to (ii). 
if 
