. The principles of projective geometry applied to the straight line and conic . is constant, A and B describe projective ranges, and if A is given, B isuniquely determined. If -7==^====r and the involution are given, the ranges A and B are projective, V(iriA2iiF2) and if .4 is given, B is uniquely determined. S44 Principtes of Projective Geometry 152. Extension of the Involution Property of a Conic. If thioiigh (1)11/ point 0 three chords be drawn the first two of wJiich meet a conic in A A, BB, whilinvolution X^X^, FjFo, then h{ABX,X,Y,Y the conic determines on the third an h[ABY,Y,X,X,]. OO
. The principles of projective geometry applied to the straight line and conic . is constant, A and B describe projective ranges, and if A is given, B isuniquely determined. If -7==^====r and the involution are given, the ranges A and B are projective, V(iriA2iiF2) and if .4 is given, B is uniquely determined. S44 Principtes of Projective Geometry 152. Extension of the Involution Property of a Conic. If thioiigh (1)11/ point 0 three chords be drawn the first two of wJiich meet a conic in A A, BB, whilinvolution X^X^, FjFo, then h{ABX,X,Y,Y the conic determines on the third an h[ABY,Y,X,X,]. OO X, Y, X2 Y2 K Let A B meet the base of the involution at [ABX,X,Y,Y^] by projection from B becomes {KX,Y,0)-{KY,XM).h [ABY,Y^X,X.,] by projection from A becomes {OY,X,K)-{OX,Y,K).But {OY,X,K) - (OX,YJ{) = {KX,Y,0) - () = 1 - (KY,X,0) - 1 + {KX,Y,0)= {KX,Y,0)-(KY,XoO).h {ABX,XJ\Y,\ = h {ABY,Y,X,X,}.h{ABX,XJ\Y,] ^ h{ABY,Y,X,X,} ^{X,X,Y,Y,) \/{Y,Y,XiX,) H{ABX,X,Y,Y,} = H {ABX,X,Y,Y,}. Converse of extension of the Involution Property. If A, A, B, B are points on a conic which determines an involution , Y^ Y^ ona given line, then if h{ABXiX2YiY.^ = h{^X.^i the three lines AA\ BBand X^X^Y^Y,^ are concurrent. Let A A meet XxX^ in 0 and let BB meet X^X^ in 0. h \A BX^X.,}, r,,} by projection from B becomes (Ali )) - {K Y^XM). h{ABY]Y.,XiX.^, by projection from A becomes {OYxXJ\:)-{{)..-. {KX,Y.,0)-{KYiX20) = {OY,X2K)-iOX\Y2K)= {KX\Y20)-{KYxX20). Therefore/i{/tOAiA2rir2} and AI/iOXiAaFiFg} have the same values and 0and 0 must coincide. Therefore Deductions an
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