. The principles of projective geometry applied to the straight line and conic . Projective Forms Harmonic 87 Let PC meet AB in K. On AB take Q the harmonic conjugate of K withrespect to AB. Then PQ is the required line. (3) Given four lines l^, I2, I3, h, and four points Pi, Po, P3, Pi on them suchthat the connectors of Pi, P2, Pz, P4 to C are harmonic conjugates of ^i, U-, I3, li withrespect to the connectors of Pj, P2) Aj -^4 to A and B, prove that the pencil(C PiP^PsPi) is projective with the range formed on AB by the lines ?j, I2, I3, ^4. (4) The lines joining any point 0 to four collinea
. The principles of projective geometry applied to the straight line and conic . Projective Forms Harmonic 87 Let PC meet AB in K. On AB take Q the harmonic conjugate of K withrespect to AB. Then PQ is the required line. (3) Given four lines l^, I2, I3, h, and four points Pi, Po, P3, Pi on them suchthat the connectors of Pi, P2, Pz, P4 to C are harmonic conjugates of ^i, U-, I3, li withrespect to the connectors of Pj, P2) Aj -^4 to A and B, prove that the pencil(C PiP^PsPi) is projective with the range formed on AB by the lines ?j, I2, I3, ^4. (4) The lines joining any point 0 to four collinear points A, B, C, D cut anytransversal through D in a, /3, -y, Z), respectively, and By meets 0^ in O, 0/3 meets^5 in E. Prove that if {AB, CD) = - 1 then = BB. it IS necessary to prove that -^^ : -^-, But 2 or that {ABBC)=-2. {ABCD) = {al3yD) = {ABBD)= - 1,{ABBD) = 2 and (45CZ))=-1, 2. JABBD) ^^^-{ABcnj-. (5) KLMNis a simple quadrangle and J, ^ are two points such that (A. LiVAB),{), {M. LNAB) are harmonic. Prove that: (i) {N. MKAB) is harmonic, and (ii) A, B be situated on the third diagonal of the the transversal AB meets the sides asin the figure in RSTU, {ABUR)=-\,{ABRS)=-V,{ABST)=^^_^_AS__AT â BU Bit BS~ BTTherefore the pencil (iV. KMAB) is har-monic. Since -jryy =7.^.5 C and S coincideJtSU JJib or ^^ passes through E. Similarly AB passes through of harmonic conjugates of E and F. (6) If ABCD be a quadrangle and bisects BB, then the connector is parallel to BD, or ABCD is a parallelogram. (7) If a transversal cut the sides BC, CA, AB of a triangle in /*, (^, Rrespectively, then (i) if P, Q, R be the harmonic conjugates of P, Q, li with respect to BC,CA, AB respectively, AF, BQ, CR! meet in a point, and (ii) if X, T, Z be the middle points of PF, QQ, RR respectively, X, Y, Zlie on a straight line. Since {BCFP)=-l, FB_PBâ
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
