Download PDF by David Gans: An Introduction to Non-Euclidean Geometry

By David Gans

ISBN-10: 0122748506

ISBN-13: 9780122748509

Publication by way of Gans, David

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Proof. Consider any triangle ABC (Fig. Ill, 16) and a variable point D between A and B. Let D approach A. Then ^ADC approaches ^EAC, and -$:ACD becomes arbitrarily small in value (§2, Property 15). Hence the angle-sum of triangle ADC gets arbitrarily close to the value of ^EAC + ZCAB, which is 180°. E A D B Fig. Ill, 16 Keeping in mind that the term "quadrilatéral" always means a convex quadrilateral for us and that the line joining two nonadjacent vertices of such a quadrilateral subdivides the angles at those vertices (§2, Property 10), we can easily obtain the following corollary of Theorem 42.

10. Does Theorem 46 hold in Euclidean geometry? Justify your answer. QUADRILATERALS ASSOCIATED WITH A TRIANGLE Each side of any given triangle ABC is the summit of a certain Saccheri quadrilateral having an important relation to the triangle. Consider side BC in Fig. Ill, 22, for example. Let D, E be the midpoints of the other sides, B C Fig. Ill, 22 and let F9 G, H be the projections of A, B, C on line DE. Triangles A DF, BDG are congruent by angle-angle-side (Theo. 26), and so are triangles AEF, CEH.

Prove that noncongruent similar triangles do not exist in hyperbolic geometry. ) 4. Prove that the statement negating any substitute for Postulate 5 is a fact of hyperbolic geometry. 5. Prove that no two lines in hyperbolic geometry are equidistant from one another by showing that the distance from one line to another cannot have the same value in more than two places. 6. The set of points which are at the same distance from a given line and lie on the same side of it is called an equidistant curve, and the line is called the base line of the curve.

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An Introduction to Non-Euclidean Geometry by David Gans


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