Circles: A Mathematical View - download pdf or read online

Geometry Topology

By Daniel Pedoe

This revised version of a mathematical vintage initially released in 1957 will convey to a brand new new release of scholars the joy of investigating that easiest of mathematical figures, the circle. the writer has supplemented this re-creation with a different bankruptcy designed to introduce readers to the vocabulary of circle suggestions with which the readers of 2 generations in the past have been common. Readers of Circles want basically be armed with paper, pencil, compass, and immediately facet to discover nice excitement in following the buildings and theorems. those that imagine that geometry utilizing Euclidean instruments died out with the traditional Greeks may be pleasantly stunned to benefit many fascinating effects that have been in basic terms came upon nowa days. newcomers and specialists alike will locate a lot to enlighten them in chapters facing the illustration of a circle by way of some extent in three-space, a version for non-Euclidean geometry, and the isoperimetric estate of the circle.

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Extra resources for Circles: A Mathematical View

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L, M, and N are collinear. This is surely a startling result, totally unexpected, easy to grasp, and beautiful to contemplate. In the modern foundations of geometry it is one of the fundamental theorems. If we take other points, say D on f and D' on m, the intersection CD' n C'D lies on the line already constructed. This line will not, we note, usually pass through the intersection of e and m. A P FIG. 340 xxxiv CIRCLES A case where it always does so arises when we choose our points A, A', B, B', and C, C' in a special way, so that the joins AA', BB', and CC' all pass through the same point V.

In the modern foundations of geometry it is one of the fundamental theorems. If we take other points, say D on f and D' on m, the intersection CD' n C'D lies on the line already constructed. This line will not, we note, usually pass through the intersection of e and m. A P FIG. 340 xxxiv CIRCLES A case where it always does so arises when we choose our points A, A', B, B', and C, C' in a special way, so that the joins AA', BB', and CC' all pass through the same point V. The theorem of Pappus still applies, and we obtain a figure almost like a national flag, seen in perspective.

Let B and C be the acute angles of the triangle ABC. On BC, and away from A, describe an equilateral triangle BCD. Then by the extension of Ptolemy's theorem, unless P lies on the circle Fio. BC, or PB + PC > PD, since CD = DB = BC. Therefore PA + PB + PC > PA + PD. 6 Now, unless P lies on AD, we have PA + PD > AD. Hence, unless P is at P' (the other intersection of AD with the circle BCD), we have PA + PB + PC> AD. But if P is at P', both the above inequalities become equalities; so that P'A + P'B + P'U = AD.

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