[GLASHAN] ALTERNATE NUMBERS AS GEOMETRICAL INDICES 187 
On substituting for the ratios y : X, etc., their equivalents in line- 
segment ratios, we obtain 
ABICR PRG AB BC: CA". AA’. BB .CC AE. BF. CG 
(ACH BAT EB i CEE AT Co BEAN: 
(AB. A‘C..CC’. AE—BC..AC’. AA’. CG) 
(BC; BA . AA’. BF=CA » BAY BB’. AE) 
(CA. CB: BB’ 2CG—AB...CB CC! . BF) |. 
Gorollary.. lf}. ACL BA CE CB. AC: B'A=0° > then: will 
A” B"C"” =0, i.e., if AE, BF and CG are concurrent, A”, B” and C” 
are collinear. 
Conversely, if A”B”’C”=0 and neither ABC=0 nor EFG=0, 
then must AC’. BA’. CB’—C'B.A’'C.B’'A=0, i.e., if A”, B” and C” 
are collinear, then will AE, BF and CG be concurrent. 
DEFINITIONS.—If the triangles ABC, EFG are in plane perspec- 
tive, and on the connectors AE, BF, CG third points O, P, Q be taken 
the triangles ABC, EFG, OPQ will be in concentric perspective. 
If the triangles abe, efg are in plane perspective and through the 
axial points À”, B”, C” third lines 0, p, q be drawn, the triangles 
abc, efg, opq will be in coaxial perspective. 
XV. If three triangles are in concentric perspective their axes of 
perspective are concurrent. 
Let ABC, EFG and OPQ be in concentric perspective, D their 
centre of perspective, A”=BC .FG, B”’=CA.GE, C’=AB. EF, 
E’=FG .PQ, F’=GE.QO, G’=EF.OP, O0”=PQ.BC, P’=QO. 
CA, Q”’=OP. AB and R=A’B’” . O’P” also let 
Aer + pert ves 
: A+ uty 
de (A —D)e+ pect ves 
| M iti 
ad FE = a+ (u—m)etve; 
N+u-—m+y 
Net wet (y—N)e 
Naa — 72 
TO: (À — D) + weet ves 
ND PL 
ind D = 
ind G = 
