[cLAsHAN] ALTERNATE NUMBERS AS GEOMETRICAL INDICES 193 
_ Qv=sWat(v—W)nes 
ind N 
h\y— oY +(v—p)u 
then will 
erotic tain a0 
2A, 
ind FG = (wax) m+ bXy —W) m= (UT x) M3 
24» 
ind IN = 
= anv(u—x) WW) mt bru x) v=o) et euly—) Qu — ox) 18 
1 2A; 
ind B’C’ ind FG ind IN — 
se es Au abCmn2ns 
He TOW) —X(u— x) 8 A1A2A3 
px) (ev) v(u—x) Qv—-oy)  a(—Y) Qu—ox) 
On evaluation the determinant factor vanishes and therefore 
B'C’, FG and IN are concurrent. 
CorOLLARY. By homology A’C’, EG and LK are concurrent as 
also are A’B’, EF and HM; hence the Ne A’B’C’, EFG and 
OPQ are coaxial. 
XX. If the triangles ABC, EFG are in plane perspective, with 
A’B’ as axis, and if B’=AC . BF, C’=AB. GF, F’=EG. BF and 
G’=EF.CG, B’C’, F’G’ and A’B’ will be concurrent. 

If Re = Xe + ves 
À + 
Ce Xe + ue 
À +u 
ind A” = ME — NE3 
Te 
Pi Wer 
n—l 
earls Steet = enr me eam re 
(A+ut+y)o+uln 
and de ee ee sede UT 
Atutv)r+vlm 
