A first course in projective geometry . his condition, these points are let EF meet BC in D. Then AF. BD. CE = - FB . DC. EA by the AF . BD . CE = - FB. DC. EA by hypothesis. Bp_DC_BD + DC BD ~ ~ DC BD-fDC .. D and D coincide. 1 in each case. METRICAL RELATIONS 27 Ex. 1. Prove that the sides of tlie triangle formed hy joining thepoints of contact of the inscribed circle of a triangle with the sides,intersect the sides in three collinear points. Ex. 2. Prove that the sides of the pedal triangle intersect those ofthe given triangle in collinear points. Ex. 3. Examine
A first course in projective geometry . his condition, these points are let EF meet BC in D. Then AF. BD. CE = - FB . DC. EA by the AF . BD . CE = - FB. DC. EA by hypothesis. Bp_DC_BD + DC BD ~ ~ DC BD-fDC .. D and D coincide. 1 in each case. METRICAL RELATIONS 27 Ex. 1. Prove that the sides of tlie triangle formed hy joining thepoints of contact of the inscribed circle of a triangle with the sides,intersect the sides in three collinear points. Ex. 2. Prove that the sides of the pedal triangle intersect those ofthe given triangle in collinear points. Ex. 3. Examine the case in which the transversal is parallel to aside of the triangle. § 5. A Theorem of Desargues. If the joins of corresponding vertices of two coplanar trianglesare concurrent, the intersections of cmrespoTuUng sides are collinear,aiul conversely. [In § 4 of the preceding chapter, this theorem was establishedfor triangles in space. It will now be proved by the methodof transversals to hold good when the triangles are coplanar.]. Fig. 10. Since BCX is a transversal cutting the sides of thetriangle BCO, bX CC 0B_ xc CO ? Fb~ ~ 28 PROJECTIVE GEOMETRY Similarl} from the triangle CAO, CY AA OC _YA A^ C^~ Af A or. AZ BB OA , and from ABO, ^B BO * A^= 1 From the product of these, it appears that BX CY AZ _XC YA ? ZB~ ~ ? .*. by the converse of Menelaus theorem X, Y, Z are collinear. Conversely. If X, Y, Z are collinear, AA BB, CC are con-current. For consider the triangles BBZ, CCY. The lines joining corresponding vertices are concurrent. The intersections of corresponding sides are thereforecollinear. /. o. A, A are collinear; AA, BB, CO meet in a point. § 6. Perspective. Two triangles possessing the above property are said to bein Plane Perspective. The point of concurrence of joins of corresponding points iscalled the Centre of Perspective. The line of collinearity of the intersections of correspondingsides is called the Axis of Perspective. In Chapter II. it was seen t
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