192 THE ROYAL SOCIETY OF CANADA 
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Eliminating x, 
—Ad(u—x) V—VatA—>) (ur—xY) (uetves) 
(A—¢) (uty) (ur—xY) —Ad(u—x) (v—Y) 
The substitution (e€3) (uv) (xy) which transforms ind H into ind L 
and ind M into ind K does not affect ind A, ind D or ind O, 
ind O=ind{AD . HM}=ind{AD . LK}, 
AD, HM and KL are concurrent at ©. 
ind O = 
Similarly it may be proved that BD, HM and IN are concurrent 
at say P and that CD, IN and KL are concurrent at say Q; hence 
OPQ is in plane perspective with ABC with D as the centre of per- 
spective; therefore ABC=abc, EFG=efg and OPQ=opq are in 
concentric perspective. 
XVIII. If the triangles ABC, EFG are in plane perspective and 
AE, BF, CG are their central connectors, the points AF . EB, BG . FC, 
CE . GA are the angular points of a triangle coaxial with the triangles 
ABC, EFG. 
The proof follows the lines of the proof of Prop. XVII of which 
this proposition is the dual. 
XIX. If the triangles ABC, EFG are in plane perspective and 
B’/=AC .BF, C’=AB.CG, I=AC.EF and N=AB. EG, then 
will FG, B’C’ and IN be concurrent. 
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