By C. G. Gibson

ISBN-10: 0521834481

ISBN-13: 9780521834483

This creation to the geometry of strains and conics within the Euclidean airplane is example-based and self-contained, assuming just a uncomplicated grounding in linear algebra. together with a variety of illustrations and several other hundred labored examples and routines, the e-book is perfect to be used as a direction textual content for undergraduates in arithmetic, or for postgraduates within the engineering and actual sciences.

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Additional resources for Elementary Euclidean geometry: An undergraduate introduction

Sample text

By the intersection property, the intersections of C, D coincide with those of C or D with the line L. The results of the previous section show that a circle intersects a line in two distinct points, just one point, or not at all. We can therefore conclude that any two distinct circles intersect in two distinct points, just one point, or not at all. In particular, the circles 28 Circles C C C D D D Fig. 2. Three ways in which circles can intersect intersect in a single point exactly when L touches C, D at the same point: in that case we say that C, D touch at that point.

7) Provided C, D have distinct centres, L is a line, known as the radical axis of the circles. 8 shows that any two distinct circles in the pencil λC +µD have the same radical axis L. By the intersection property, the intersections of C, D coincide with those of C or D with the line L. The results of the previous section show that a circle intersects a line in two distinct points, just one point, or not at all. We can therefore conclude that any two distinct circles intersect in two distinct points, just one point, or not at all.

Have the same centre. In that case α = α , β = β , and L(x, y) = γ − γ is a non-zero constant function: it follows that distinct concentric circles do not intersect. However obvious that fact may be visually, it does require formal proof! 5 The radical axis of the circles displayed below is the line x = 0. Substituting x = 0 in the first circle gives y 2 + 1 = 0, having no solutions. The radical axis does not therefore meet the first circle, and the circles do not intersect x 2 + y 2 − 3x + 1 = 0, 2x 2 + 2y 2 − 7x + 2 = 0.

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