Translation of Fuhrmann’s “Sur un nouveau cercle associé à un triangle”

We provide a translation (from the French) of Wilhelm Fuhrmann’s influential 1890 paper in which he introduced what is today called the Fuhrmann triangle. We have added footnotes that indicate current terminology and references, ten diagrams to assist the reader, and an appendix. Introductory remarks Current widespread interest concerning the Fuhrmann triangle motivated us to learn what, exactly, Fuhrmann did in his influential 1890 article, “Sur un nouveau cercle associé à un triangle” [5]. We were quite surprised to see how much he accomplished in that paper. It was clear to us that anybody who has an interest in triangle geometry might appreciate studying the work for himself. We provide here a faithful translation of Fuhrmann’s article (originally in French), supplementing it with footnotes that indicate current terminology and references. Two of the footnotes, however, (numbers 2 and 12) were provided in the original article by Joseph Neuberg (1840-1926), the cofounder (in 1881) of Mathesis and its first editor. Instead of reproducing the single diagram from the original paper, we have included ten figures showing that portion of the configuration relevant to the task at hand. At the end of our translation we have added an appendix containing further comments. Most of the background results that Fuhrmann assumed to be known can be found in classic geometry texts such as [7]. Biographical sketch. Wilhelm Ferdinand Fuhrmann was born on February 28, 1833 in Burg bei Magdeburg. Although he was first attracted to a nautical career, he soon yielded to his passion for science, graduating from the Altstädtischen Gymnasium in Königsberg in 1853, then studying mathematics and physics at the University of Königsberg, from which he graduated in 1860. From then until his Publication Date: February 18, 2010. Communicating Editor: Paul Yiu. Translated from Franz Kössler, Personenlexikon von Lehrern des 19 Jahrhunderts: Berufsbiographien aus Schul-Jahresberichten und Schulprogrammen 1825-1918 mit Veröffentlichungsverzeichnissen. We thank Professor Rudolf Fritsch of the Ludwig-Maximilians-University of Munich and his contacts on the genealogy chat line, Hans Christoph Surkau in particular, for providing us with this reference. 14 J. Vonk and J. C. Fisher death 44 years later he taught at the Oberrealschule auf der Burg in Königsberg, receiving the title distinguished teacher (Oberlehrer) in 1875 and professor in 1887. In 1894 he obtained the Order of the Red Eagle, IV Class. He died in Königsberg on June 11, 1904. On a New Circle Associated with a Triangle Wilhelm Fuhrmann 2 Preliminaries. Let ABC be a triangle; O its circumcenter; I, r its incenter and inradius; Ia, Ib, Ic its excenters; Ha,Hb,Hc,H the feet of its altitudes and the orthocenter; A, B, C ′ the midpoints of its sides; ABC ′′ the triangle whose sides pass through the vertices of ABC and are parallel to the opposite sides.