Topics in Elementary Geometry

This small book, now translated into English, is a unique place to find classical results from geometry. The selection of topics is intelligent, varied, and stimulating. In a small space the author provides many thought-provoking ideas.

Oene Bottema (1901-1992) may not be so well known abroad, but in his own country he is "the great geometer". He graduated from the University of Groningen in 1924 and obtained his doctor's degree from Leiden University in 1927. He spent his early years as a high school teacher and administrator. He published extensively, and as his ability became known, he was made professor at the Technical University of Delft in 1941, and later rector of thatuniversity(1951-1959). Withhisencyclopedicknowledgeof19th-century geometry and his training in 20th-century rigor, he was able to make many contributions to elementary geometry, even as that subject was eclipsed by the modern emphasis on abstract mathematical structures. He also had a fruitful collaboration with engineers and made substantial contributions to kinematics, culminating in the book Theoretical Kinematics, withBernard Roth, in 1979. Throughout his life he was inspired by geometry and poetry, and favored elegant succinct proofs. This little book, ? rst published in 1944, then in a secondexpanded edition in 1987, gives us a glimpse into his way of thinking. It is a series of vignettes, each crafted with elegance and economy. See, for example, his proof of the Pythagorean theorem (1. 2), which requires only one additional line to be drawn. And who can imagine a simpler proof of the nine-point circle (4. 1)? There is ample coverage of the modern geometry of the triangle: the Simson line, Morley's theorem, isogonal conjugates, the symmedian point, and so forth.

The Pythagorean Theorem. - Ceva#x02019; s Theorem. - Perpendicular Bisectors; Concurrence. - The Nine-Point Circle and Euler Line. - The Taylor Circle. - Coordinate Systems with Respect to a Triangle. - The Area of a Triangle as a Function of the Barycentric Coordinates of Its Vertices. - The Distances from a Point to the Vertices of a Triangle. - The Simson Line. - Morley#x02019; s Triangle. - Inequalities in a Triangle. - The Mixed Area of Two Parallel Polygons. - The Isoperimetric Inequality. - Poncelet Polygons. - A Closure Problem for Triangles. - A Class of Special Triangles. - Two Unusual Conditions for a Triangle. - A Counterpart for the Euler Line. - Menelaus#x02019; s Theorem; Cross-Ratios and Reciprocation. - The Theorems of Desargues, Pappus, and Pascal. - Inversion. - The Theorems of Ptolemy and Casey. - Pedal Triangles; Brocard Points. - Isogonal Conjugation; the Symmedian Point. - Isotomic Conjugation. - Triangles with Two Equal Angle Bisectors. - The Inscribed Triangle with the Smallest Perimeter; the Fermat Point.