By O. Bottema

This small booklet has for a very long time been a special position to discover classical effects from geometry, akin to Pythagoras' theorem, the nine-point circle, Morley's triangle, Poncelet's polygons, and lots of different topics. furthermore, this ebook comprises fresh, geometric theorems which were received during this classical box during the last years. There are 27 autonomous chapters on a variety of issues in undemanding aircraft Euclidean geometry, at a degree simply past what's often taught in a superb highschool or university geometry path. the choice of themes is clever, different, and stimulating. In a small area the writer presents many thought-provoking principles. This ebook will slot in good with the expanding curiosity for geometry in study and schooling. This e-book was once initially released in Dutch, and it will be the 1st English translation. This translation additionally incorporates a new foreword by means of Robin Hartshorne.

"This hugely interesting booklet will increase the reader's historic viewpoint in an enlightening demeanour and it presents appealing subject matters for school room discussion." -Hendrik Lenstra, Universiteit Leiden

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**Sample text**

1 In the preceding chapters, our tools have been primarily “geometric”: congruence, similarity, inscribed angles, trigonometric functions. We can also take an “analytic” approach to geometry, representing points by coordinates, and lines and curves by equations. The most commonly used coordinate systems are the Cartesian and polar coordinates. The most important ﬁgure in elementary plane geometry is the triangle, and much of what is treated in this book relates to it. Because of its importance, there exist coordinate systems that are associated to the triangle that is being considered.

4) if and only if ap, bq, and cr satisfy the triangle inequalities. 3 By what we have just seen, the distances r1 , r2 , and r3 from a point P to the vertices of a triangle ABC are unsuitable for use as a coordinate system. Their ratios cannot be used either: they correspond to two, one, or zero real points. They are no true rival for the trilinear or barycentric coordinates. However, the elaborate computations of this chapter have not been completely in vain. There are questions of geometric type for which they can be used.

We can now apply the construction to an arbitrary convex ﬁgure (Fig. 6). Imagine all chords P P perpendicular to l as translated in their own direction until their midpoints lie on l. The area has not changed, on account of Cavalieri’s principle, while we can prove that 62 13 The Isoperimetric Inequality P P P2 l P1 Fig. 6. the perimeter has not increased. By cutting the ﬁgure up into “small” strips using lines perpendicular to l, and as such approximating the ﬁgure with a polygon, we facilitate our acceptance of these results.