. The principles of projective geometry applied to the straight line and conic . ined, viz. If through the point of inter-section of the tangents at any pointsT and T on a circle, a chord bedrawn to meet the circle in A andA, then the range TTAA isharmonic. If from any point on the choi^dof contact of any tiuo tangents t andt to a circle, tiuo other tangentsa and a be drawn to the circle,then the pencil of tangents ttaais harmonic. 130 PrmripJes of Projective Geometry Conversely : If A A, BB, CC are pairs of con-jugate points of an involution on a circle,the lines AA\ BB, CC are concurrent. Th
. The principles of projective geometry applied to the straight line and conic . ined, viz. If through the point of inter-section of the tangents at any pointsT and T on a circle, a chord bedrawn to meet the circle in A andA, then the range TTAA isharmonic. If from any point on the choi^dof contact of any tiuo tangents t andt to a circle, tiuo other tangentsa and a be drawn to the circle,then the pencil of tangents ttaais harmonic. 130 PrmripJes of Projective Geometry Conversely : If A A, BB, CC are pairs of con-jugate points of an involution on a circle,the lines AA\ BB, CC are concurrent. The involution is determined by twopairs of conjugate points AA\ A A, BB meet at S. Join C to ,Sfto meet the circle in C. Then C isa conjugate of C in the given involutionand must therefore coincide with €. Hence CC passes through S. 76. Pole and Polar. If through any fixed pointa variable chord of a circle hedrawn, the locus of the harmonicconjugates of the fixed point ivitJtregard to the points of intersectionof the variable chord loith the circleis a, straight Draw through 8 three chordsmeeting the circle in AA, BB,CC The first two chords maybe regarded as fixed and the lastas variable. The six points forman involution on the circle andtherefore the pencil formed byjoining A, A, B, B, C, C to A is If aa, bh\ cc are pairs of conjugaterays of an involution pencil of tangentsto a circle, the points aa, hh, cc arccollincar. The involution is determined by twopairs of conjugate tangents ao^, hh. Letthe connector of aa to hh be s. T^etc meet s in C and let c be the tangentfrom C to the circle. Then c is a con-jugate of c in the given involution andmust therefore coincide with c. Hence cc lies on s. If from a variable point ona fixed straight line pairs of tan-gents be drawn to a circle, theenvelope of the harmonic conjugateof the fixed line with regard to thepair of variable tangents is a. fixedpoint.
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
