By I. M. Yaglom, Abe Shenitzer
This booklet is the sequel to Geometric Transfrmations I and II, volumes eight and 21 during this sequence, yet should be studied independently. it truly is dedicated to the remedy of affine and projective variations of the airplane; those alterations contain the congruences and similarities investigated within the earlier volumes. the straightforward textual content and the various difficulties are designed typically to teach how the priniciples of affine and projective geometry can be utilized to provide particularly uncomplicated suggestions of huge periods of difficulties in simple geometry, together with a few directly part building difficulties. within the complement, the reader is brought to hyperbolic geometry. The latter a part of the booklet comprises particular ideas of the issues posed in the course of the textual content.
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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.