The Geometry of Remarkable Elements: Points, Lines and Circles

by Constantin Mihalescu, Titu Andreescu

This book is an English translation of a text written by Constantin Mihalescu, a retired artillery colonel and enthusiastic amateur mathematician. With the majority of the results obtained in the second half of the 19th century and the first half of the 20th century, this book was one of the most complete descriptions of geometry of its time. It contains a comprehensive collection of the most important properties of points, lines, and circles related to triangles and quadrilaterals, as they were known by the mid-1950s, and a rich assortment of problems to entice and inspire readers of all levels. Topics covered include the nine-point circle, the Simson line, the orthopolar triangles, the orthopole, the Gergonne and Nagel points, the Miquel point and circle, the Carnot circle, the Brocard points, the Lemoine point and circles, the Newton-Gauss line, and much more.

Genres: Mathematics

577 pages, Hardcover

Published May 15, 2016

Reviews

[Tom Schulte · Rating 4 out of 5]

...Skipping past Euclid, Pythagoras, Thales, and other well-known geometers of antiquity, this impressive collection launches into the Euler’s nine-point circle whose circumference features a nonet of concyclic points defined from the triangle. This is explored for over one hundred pages yielding a bounty of features and attributes: homothety, the pedal triangle, Mathot’s point, and much more. Fortunately, this detailed taxonomy of the concurrent, cyclic, and collinear is exemplified in a profusion of illustrations. The book is dense with graphics artfully done: black-lined main figures with emanating and encircling additions rendered in pastel green, red, and light purple. Each is ready to be enlarged and displayed in a corporate lobby; a destiny I particularly recommend for Figure 56 tracing “…the isotomic (reciprocal) transversals of the tangents of the nine-point circle…”...

[Look for my complete review in MAA Reviews.]