. The principles of projective geometry applied to the straight line and conic . on s. The correlative theorem, which may be proved by similar methods,is as follows. If five paiis of vertices of two quadrilaterals are collinear with a givenpoint, then the sixth pair are also collinear with this point. Projective Forms Anharmonic 21 From this it follows that if two quadrangles ABCD and ABCD are suchthat five of the sides of one are parallel to the five corresponding sides of the other,then the sixth pair of sides are also parallel. The axis of Perspective is the line atinfinity. (c) Cevas Theor
. The principles of projective geometry applied to the straight line and conic . on s. The correlative theorem, which may be proved by similar methods,is as follows. If five paiis of vertices of two quadrilaterals are collinear with a givenpoint, then the sixth pair are also collinear with this point. Projective Forms Anharmonic 21 From this it follows that if two quadrangles ABCD and ABCD are suchthat five of the sides of one are parallel to the five corresponding sides of the other,then the sixth pair of sides are also parallel. The axis of Perspective is the line atinfinity. (c) Cevas Theorem. If the lines joining any point P to the verticesABC of a triangle meet the opposite sides in ABC respectively, thenBA CB AC _CA AB- BG~ Draw through B and C lines parallelto AA to meet CA and BA respectivelyin B and C. In this case x , the pointat infinity on AA, is also the point atinfinity on BB and CO. Then {) = {),.-. {AGBB) = {ABCC),by taking intercepts on AC and AB. AB AB _A(r AC^• • CB- CB~ BC BC AB BC _AB AG^■ ■ CBAC~ CBbC AB BC CA __. GBACBA~. AB ACCB BC BAf_ CA Conversely. If points ABC are taken on the sides of a triangle ABC such thatBA CB AC CAABBC = -1, then the lines A A, BB, CC are concurrent. Let the lines BB, CC intersect in P, and let AP meet BC in A. Then j5^ C^ AC_CA AB BC A »• vni ^ BA ^ BA CA CA . A coincides with A. - (from data), (d) Menelaus Theorem. If ctiiy straight line meets the sides of atriangle ABC in points ABC, then BA CB AC CA • AB BC = 1, 22 Principles of Projective Geometry Let the sides of the triangle bea, h, c and let the straight line be A draw b, parallel to b, tomeet c in C. Let AA be a and let the point atinfinity on 6 be oo . Then {b. aapb) ■-= (c. aapb),.-. {CABzo) = {BACC)by taking intercepts on 6 and c. CB^mr BG ■ ■ AB~ AC ACCEABGBAB
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
