. The principles of projective geometry applied to the straight line and conic . If SL and SM the tangents from S meet q in L and J/, and L, M be takensuch that LL = LL and MM = MM\ then SL and SM touch the envelope at Land M, and its centre € is the point of intersection of the perpendiculars throughM and L. (22) To find the centre and axis of perspective of two circles one of which issituated entirely within the other. Through C and C the centres of thecircles, whose radii are r and ?•, draw anypair of parallel lines to meet the circles inP and P and let PP meet CC in S andthe radical axis i
. The principles of projective geometry applied to the straight line and conic . If SL and SM the tangents from S meet q in L and J/, and L, M be takensuch that LL = LL and MM = MM\ then SL and SM touch the envelope at Land M, and its centre € is the point of intersection of the perpendiculars throughM and L. (22) To find the centre and axis of perspective of two circles one of which issituated entirely within the other. Through C and C the centres of thecircles, whose radii are r and ?•, draw anypair of parallel lines to meet the circles inP and P and let PP meet CC in S andthe radical axis in R. ^, CS PS CP r „. -Then ^ = p^. = ^7^= / • Therefore S is a centre of similitude of the circles and is a fixed point. (Art. 84.) (1) Take .S as centre and the line at infinity as axis of pei-spective. SP Then {Sao PP)-- SP Hence the. circles are in perspective. (2) Let SPP meet the circles again in Q and (/. Then S(lSQ Take S as centre and the radical axis as axis of _RP^^ PP _SP -SP RQ~ Rq~ qq~ sq-sq SP_ RP^SP SP-SP SQ-nq sq-SQ-sq Therefore the circles are again in perspective. A second centre of perspective »S may be found by producing CP to meet thelarger circle in Px and determining S from /j and P. Then But {SRPq)-- ^e->> CHAPTER XII PROJECTIVE THEOREMS FOR THE CIRCLE :—CARNOTS THEOREM. DESARGUES THEOREM 89. The proofs of Carnots, Pascals, Desargues Theorems, andtheir correlatives for the circle are very similar to those for the conic,but in some cases they take a slightly different form and the proofs aresimpler. When proved for the circle they may be deduced for theconic by projection. In this chapter, certain proofs for the circle aregiven, but the full discussion of these theorems is left over till they areconsidered for the conic in Chapter xiv. Carnots theorem for
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