Download Topics in Geometry by Robert Bix PDF
By Robert Bix
This quantity offers an obtainable, self-contained survey of themes in Euclidean and non-Euclidean geometry. It contains abundant illustrations and routines in aid of the completely worked-out proofs. The author's emphasis at the connections among Euclidean and non-Euclidean geometry unifies the diversity of themes covered.
The textual content opens with a quick assessment of effortless geometry earlier than continuing to complex fabric. issues coated comprise complicated Euclidean and non-Euclidean geometry, department ratios and triangles, transformation geometry, projective geometry, conic sections, and hyperbolic and absolute geometry. Topics in Geometry contains over 800 illustrations and large workouts of various trouble
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Additional info for Topics in Geometry
Sample text
Let A' be a point on BC other than B and C, and let B' be a point on CA other than C and A. Assume that AB and A'B' aren't parallel. 7). Proof: A' doesn't equal the point C where CA and BC intersect, and so A' doesn't lie on CA. ) It follows that A'B' ^ CA, and so these lines intersect at the unique point B'. Thus, A'B' doesn't contain A. Then A'B' doesn't contain B either, by symmetry. Since AB and A'B' aren't parallel, by assumption, they intersect at a unique point C. We've seen that neither A nor B lies on A'B', and so neither of these points equals C.
In the Euclidean plane, let ABC be a triangle whose sides have lengths \AB \ = 8, | 5 C | = 3, and \AC\ = 6. Let £ ' be the point on line AC that lies 4 units from A between A and C, and let C" be the point on line AB that lies 2 units from A between A and 5 . Prove that the lines BC and B'C intersect at a point A', and find the distances from A' to 1? and C. Illustrate your answer with a figure. 4. In the Euclidean plane, let ABC be a triangle whose sides have lengths | AB | = 5, \BC\ = 4, and |^4C| = 2.
THEOREM 2 . 2 . In the Euclidean plane, let ABC be a triangle. Let A' be a point on BC other than B and C, and let B' be a point on CA other than C and A. 4). 12 shows that AB and A'B' are parallel if and only if the equality A'B/A'C = B'A/B'C (4) holds. 2i), we can obtain Equation 4 by multiplying Equation 3 by B'A/B'C, and we can obtain Equation 3 by multiplying Equation 4 by B'C/B'A. Thus, AB and A'B' are parallel if and only if Equation 3 holds. 2 gives a criterion for AB and A'B' to be parallel.