[cLAsHAN] ALTERNATE NUMBERS AS GEOMETRICAL INDICES 189 
On eliminating x, we obtain 
dR= ld(my —nx)a+mx(nd —lp)e+ny(lx — mp}es 
(l—n) (mp—nx)o—(m—n) (ly—nd)x 
ind R ind E” ind F’= 
IL mxM, ny Ny A 
ALi ply+mx(n—¥) vL,—ny(m— x) | 916263 
AM, —lo(n—-V) uM, vM,+ny(l— ¢) 
in which 
L,=my—nx, M=no—ly, Ni=lx—me¢, 
à =/6L,+myxM+nyN, 
bs =(Atutv—m)Li+(m—n) (m—x)Y 
ds =(A+ut+tv—n)Mi+(n—l) (n—p)¢. 
On evaluating the determinant it will be found to vanish, hence 
ind R ind E” ind F”=0, and therefore the line E’F” passes through 
the point R, i.e., the axial lines A”B”, E’F” and O’P” are concurrent. 
CoROLLARY.—Hence, if three triangles are in concentric perspec- 
tive, their axial points are the angular points of three other triangles 
also in concentric perspective, and the axes of perspective of either 
triad are the central connectors of the other triad. 
XVI. If three triangles are in coaxial perspective, their centres 
of perspective are collinear. 
Let ABC, EFG, OPQ be in coaxial perspective and let D, be the 
centre of perspective of ABC and EFG, D, the centre of ABC 
and OPQ, D; the centre of EFG and OPQ, A’=BC .FG. PQ, 
"=CA. GE. QO and C’”=AB.EF. OP, also let 
D Aer + Mest vez 
re am 
ME (À —l)e + ue +ves 
Neal 
‘de Ne + (u — m1) e+ Ves 
Atutr-m 
ind QE Aa + ue +(r—n)es 
ÀAtu+r-n 
