What became of Karl Feuerbach? Geometers universally regard the so-called Feuerbach theorem as undoubtedly one of the most beautiful theorems in the modern geometry of the triangle. This theorem concerns itself with 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).* Now the Feuerbach theorem says that for any triangle, the nine-point circle is tangent to the incircle and to each of the three encircles of the triangle.
The theorem was first stated and proved by Karl Wilhelm Feuerbach (1800-1834) in a little work of his published in 1822. It constitutes his only claim to fame in the field of mathematics~Why did he not produce further? What became of him? Why did he die at so young an age as thirty-four ? The answers to these questions constitute quite a tale.
Karl, the third son in a family of eleven children, was born in Jena on May 30, 1800. His father was a famous German jurist, becoming in 1819 the president of the court of appeals in Ansbach. Karl studied at both the University of Erlangen and the University of Freiburg, and in 1822 published his little book containing the beautiful theorem. He
During the incarceration, Karl became obsessed with the idea that only his death could free his companions. He accordingly one day slashed the veins in his feet, but before he bled to death he was discovered and removed in an unconscious state to a hospital. There, one day, he managed to bolt down a corridor and leap out of a window. But he fell into a deep snowbank and thus failed to take his life, though he did emerge permanently crippled so that later he looked like a walking question mark.
Shortly after his hospital adventure, Karl was paroled in the custody of a former teacher and friend of the family. One of the other nineteen young men died while in prison, and it was not until after fourteen months that a trial was held and the men were vindicated and released. King Maximilian Joseph took great pains to assist the young men in returning to normal life.
Karl was appointed professor of mathematics at the Gymnasium at Hof, but before long he suffered a breakdown and was forced to give up his teaching. By 1828 he recovered sufficiently to resume teaching, this time at the Gymnasium at Erlangen. However, one day 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. This wild and unbecoming act earned him permanent retirement. He gradually withdrew from reality, allowed his hair, beard, and nails to grow long, and became reduced to a condition of vacant stare and low unintelligible mumbling. After living in retirement in Erlangen for six years, he quietly died on March 12, 1834.