By Kasner E.
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Example text
P. 18) in that here x denotes a projective plane rather than an ordinary plane. plane x $ This assumption does not restrict the generality of the theorem: In fact, in any region G i t is possible to choose a (small) quadrilateral A B C D . Every line intersecting A B C D must pass through G. But then every transformation which carries lines passing through G into lines must carry lines intersecting the (convex) quadrilateral A B C D into lines. PROJECTIVE TRANSFORMATIONS 53 and so we merely sketch the required argument.
54 GEOMETRIC TRANSFORMATIONS f Figure 51 3. Central projections which carry a circle into a circle. Stereographic projection. I n the preceding section we stated a number of problems whose solutions were simplified by means of an appropriate projection of the plane of the figure to another plane. It is natural to consider the scope of this method. I n elementary geometry one studies the properties of figures made up of lines and circles. Central projections preserve lines, but not, in general, circles.
Divide the segment A B into n equal parts by means of straightedge alone. Figure 42 33. What form will the theorem on the complete quadrilateral (cf. Problem 14 and the comment which follows it) take if we project Fig. 14 so that line A B E is the special line? 34. (a) Prove the theorem of Menelaus: Three points M , N , P on sides A B , BC and C A of a triangle ABC (or on their extensions, see Fig. - AM BN CP = 1. BM CN A P (b) Prove the theorem of Ceoa: Three lines A N , B P and C M , where the points M , N , P lie on the sides A B , BC, C A of A A B C (or on their extensions, see Fig.