. The principles of projective geometry applied to the straight line and conic . versal AB meets the sides asin the figure in RSTU, {ABUR)=-\,{ABRS)=-V,{ABST)=^^_^_AS__AT ■ 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
. The principles of projective geometry applied to the straight line and conic . versal AB meets the sides asin the figure in RSTU, {ABUR)=-\,{ABRS)=-V,{ABST)=^^_^_AS__AT ■ 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_PB•PXI~ PC lience A, B arc any 88 Principles of Projective Geometnj and Similarly qA QAEA^RAEB~ • • PC. QA. RB ~ ^ PC. QA. RB= -1 by Menelaus by Cevas TheoremAP, Bq. CR are fPB ^^^ Since XC FBy .^ PC) XB. follows , (^^^ XC. ^ \ by Menelaus Theorem X, Y, Z are collinear. Given that AP, B^, CR are concurrent, the converse theorem can be point of concurrency of AP, BQ, CR is termed the pole of the linePQR with respect to the triangle. This result should be compared with Example 11, Chapter VI. (8) The middle points of the diagonalsof a quadrilateral are collinear. In the figure the sides of the triangleABC are cut by a transversal PQR andfrom the quadrangle QQPP, PP, QQ,RR divide the sides of ABC the middle points of PP, QQ, RRare collinear. But PP, QQ, RR are thediagonals of the quadrilateral the theorem is true. (9) The three straight lines joining the vertices A, B, C of a, triangle to a point cut the opposite sides in points P, Q, Rrespectively, and R is the point on BA which divides it externally in t
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