By Daniel Pedoe

This revised variation 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 distinct bankruptcy designed to introduce readers to the vocabulary of circle suggestions with which the readers of 2 generations in the past have been known. Readers of Circles desire in basic terms be armed with paper, pencil, compass, and instantly facet to discover nice excitement in following the structures and theorems. those that imagine that geometry utilizing Euclidean instruments died out with the traditional Greeks might be pleasantly stunned to benefit many fascinating effects which have been in basic terms found nowa days. rookies 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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81 (1955) pp. 71-76. 4. Feuerbach, K. , Eigenschaften einiger merkwurdiger Punkte des gradlinigen Dreiecks, Nurnberg, 1822. 5. Brianchon, C. J. and Poncelet, J. , Gergonne'sAnnales de Mathematique, 11 (1821) p. 205. 6. Pedoe, Dan, "The Principle of Duality," Mathematics Magazine, vol. 48, no. 5 (1975) pp. 274-277. CHAPTER I properties of circles discussed in this chapter are those which have the habit of appearing in other branches of mathematics. THE 1. The nine-point circle This circle is the first really exciting one to appear in any course on elementary geometry.

This should be verified. Another theorem of importance is that any two circles of a coaxal system have the same radical axis. This explains the term coaxal system. We can verify this directly. liC'_ _ AC A2f + IL AS-+ A2c ± itsc A - 0. This easily reduces to (Alth- A211)[2X(g - g') + 2y(f - f') + c - c'] = 0. It follows that the centres of all circles of a coaxal system lie on a line perpendicular to the radical axis. This is also clear from the fact that the centre of AC + Fat = 0 is the point A(- f) + [L-f [A(-g) + t±(- g') l A> A+> ] In the next chapter we shall need to be familiar with a few more properties of coaxal systems, and we derive them here.

The centres of similitude of two circles FMG. 17 FMG. 18 These points have already appeared in some of our proofs. We now define them. External centre of similitude. Let W and IF' be two circles of unequal radii, with centres C and C'. Let CP and C'P' be parallel radii drawn in the same sense. Then it is clear that P'P cuts U'C in a fixed point 0, which divides U'C externally in the ratio of the radii of the two circles. This is the external centre of similitude. If the ratio of the radii = k, then OP': OP = k.