The Geometry of Remarkable Elements: Points, Lines, and Circles

This wonderful compendium, a triumph of triangle exegesis, is by a retired artillery colonel and avid mathematics enthusiast. Compiling results from a century ago and earlier, the work’s charm is its comprehensive curation of a dizzying gallery of special points, lines, and circles related to triangles and a few other shapes, such as quadrilaterals. As such, the title on the cover and heft of the tome suggest a broader review of geometry, while the thorough mining of those three connected points (as the author says, “in a concentrated and connected manner”) supports the adjective “remarkable”. Indeed, as the editors to this English translation of the Romanian text, “the book contains many results, which are hardly available elsewhere, some of them unknown to western readers.”

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…”

Such delving into the triangle’s secrets mark this as a work of passion — perhaps too avid and even fervent for pedagogy when considered in toto. As interesting as the nine-point circle is, it must be dispensed with in no more than a few lectures in a typical college geometry course and here we have nearly a semester’s worth of proofs and lemmas. Besides the individual reader of matched keen ardor, however, it is easy to envision a classroom of future artists and architects drinking in the Simson line, orthopolar triangles, and the Gergonne point and thus opening themselves up to a plethora of possibilities that nevertheless require little mathematical background.

Following ten chapters spread over nearly four hundred pages come two significant appendices. Appendix A “Preliminary Theorems to the ones from the First Part” is nearly 150 pages of content very much like and related to the main chapters and thus feels very much like a continuation and not adjunct reference material. Appendix B “Directed Triangles” stands alone as a concise monograph on ideas of French mathematician Michel Chasles.

This encyclopedic study highlighting the orthopole, Nagel points, the Miquel point, the Carnot circle, the Brocard points, the Lemoine point, the Newton-Gauss line, and much more could benefit from a more in-depth and flatter index than the two-and-a-half pages here, and also a glossary of technical terms. The short index does have entries for “cevians” and “symmedian”, should those definitions have slipped the reader’s mind, but “homology” is only a sub-entry below “circle” and to find the page where “antiorthic” is introduced one must know to look first under “axis”.

Tom Schulte prepares a glossary for his mathematics students at Oakland Community College in Michigan.