Geometers universally regard the "Feuerbach theorem" as a stunning theorem in the modern geometry of the triangle. This theorem concentrates on five important circles related to a triangle. These five circles are the incircle (or circle inscribed in the triangle), the three encircles (or circles touching one side of the triangle and the other two produced), and the nine-point circle (or circle passing through the three midpoints of the sides of the triangle). The Feuerbach theorem states that for any triangle, the nine-point circle is tangent to the incircle and to each of the three encircles of the triangle. Karl Wilhelm Feuerbach established this theory in a little-known work published in 1822 in his thin book, Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten Linien und Figuren: Eine analytisch-trigonometrische Abhandlung (Properties of some special points in the plane of a triangle, and various lines and figures determined by these points: an analytic-trigonometric treatment).

Feuerbach was born in Jena, Germany, on May 30, 1800 to Paul Johann Anselm Ritter von Feuerbach, a famous German jurist and legal scholar. Karl studied at both the University of Erlangen and the University of Freiburg. By the age of 22, he had been awarded his doctorate, been appointed to a professorship at the Gymnasium at Erlangen, and had published his theorem. His life, however, did not go well. His career as a teacher only lasted six years. During that time he suffered a breakdown and was forced to retire from teaching in 1828. Shortly after, he appeared in class with a drawn sword and threatened to behead any student who failed to solve some equations he had written on the blackboard.

He became a recluse, allowing his hair, beard and nails to grow. After living in Erlangen in this condition for six years, he died on March 12, 1834. Before he died, he published one more work in 1827, Grundriss zu analytischen Untersuchungen der dreyeckigen Pyramide (Foundations of the analytic theory of the triangular pyramid). In this work, Feuerbach introduced homogeneous coordinates. He must therefore be considered as the joint inventor of homogeneous coordinates since Möbius, in his work, Der barycentrische Calcul, also published in 1827, introduced homogeneous coordinates into analytic geometry.

Image: Karl Wilhelm Feuerbach.