A first course in projective geometry . ngles to EF. .. the triangles OCE, OCF are congruent, and the angleBOD is bisected by OC. It must he carefully noted, however, that a harmonic j^encil doesiwt in general possess a pair of conjugate rays at right angles. §8. Harmonic Properties of the Complete Quad-rangle and the Complete Quadrilateral. The nature of these figures has already been explained, andthe correspondence between them pointed out in Chap. II. § 10. The properties in question may be exhibited dualisticallyas follows: Either pair of the sides of thetriangle formed by the diagonalpoi
A first course in projective geometry . ngles to EF. .. the triangles OCE, OCF are congruent, and the angleBOD is bisected by OC. It must he carefully noted, however, that a harmonic j^encil doesiwt in general possess a pair of conjugate rays at right angles. §8. Harmonic Properties of the Complete Quad-rangle and the Complete Quadrilateral. The nature of these figures has already been explained, andthe correspondence between them pointed out in Chap. II. § 10. The properties in question may be exhibited dualisticallyas follows: Either pair of the sides of thetriangle formed by the diagonalpoints of the quadrangle forms aharmonic pencil ivith the pair ofojyposite sides of the quadranglewhich meet at the same diagonalpoint. Either pair of the vertices of thetriangle formed by the diagonallines of the q^utdrilateral forms aharmonic range with the pair ofopposite vertices of the quadrilateralwhich lie on the sam£ diagonalline. We shall prove the theorem for the quadrangle E, F, G be the diagonal points (Fig. 14a).. Let FG cut AB in H. Then, applying Cevas theorem to the triangle FAB, we + 38 PROJECTIVE GEOMETRY AH HB AE EB AH AE HB EB Also considering the transversal DCE cutting the sides ofthe same triangle, by Menelaus theorem BC= -DA. EB . CF. It follows that or .*. the range AH BE is harmonic. .*. the pencil obtained by joining F to these points isharmonic. Similarly the pencil formed by the lines EF, ED, EG, EAis harmonic. Also the pencil formed by joining G to AH BE is harmonic. The theorem is therefore proved for the quadrangle.
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
