Died: 14 January 1912 in Porrentruy, Switzerland

Arnold Droz-Farny adopted the name Farny later in life after he married. He was named Arnold Droz, and was the son of Edouard Droz and his wife Louise. He studied mathematics in the canton of Neufchâtel and then continued his studies in Munich where he attended analysis lectures by Klein. By this time he had developed a strong preference for geometry. He married Lina Farny who was also from La Chaux-de-Fonds. She was the daughter of Constant and Anna Farny, and was a little older than her husband being born on 26 April 1854. On marriage they both changed their name to Droz-Farny.

After studying in Munich, he was appointed as a teacher "in an important institute of German- 
speaking Switzerland". In 1880 Droz-Farny was appointed as a professor of physics and mathematics at the cantonal School in Porrentruy near Basel in Switzerland. He taught there until 1908.

Eduardo L Ortiz writes:-

All I know of his personal life is that he seems to have been very sociable, liked mountain climbing and horse races, and kept correspondence with a number of mathematicians, among them Virginio Retali in Italy, and Juan Jacobo Duran Loriga in Spain. With the latter he kept correspondence between1889 to 1910 when Duran Loriga died.

Ayme writes [4]:-

[Droz-Farny] is known to have written four books between 1897 and 1909, two of them about geometry. He also published in the Journal de Mathématiques élemantaires et Spéciales (1894, 1895), and in L'intermédiare des Mathématiciens and in the Educational Times (1899) as well as in Mathesis(1901).

Droz-Farny is best known for results published in the publications of 1899 and 1901 mentioned in this quote. The first of these was Question 14111 in The Educational Times 71 (1899), 89-90. In this he stated the following remarkable theorem without giving a proof:

If two perpendicular straight lines are drawn through the orthocentre of a triangle, they intercept a segment on each of the sidelines. The midpoints of these three segments are collinear.

This is known as the Droz-Farny line theorem, but it is not known whether Droz-Farny had a proof of the theorem. Looking at other work by Droz-Farny, one is led to conjecture that indeed he would have constructed a proof of the theorem. The 1901 paper we mentioned above is, for example, one in which he gives a proof of a theorem stated by Steiner without proof. Droz-Farny's proof appears in the paper Notes sur un théorème de Steiner in Mathesis 21 (1901), 268-271. The theorem is as follows:

If equal circles are drawn on the vertices of a triangle they cut the lines joining the midpoints of the triangle in six points. These six points lie on a circle whose centre is the orthocentre of the triangle.

Articles:

1. J-L Ayme, A purely synthetic proof of the Droz-Farny line theorem, Forum Geom. 4 (2004), 219-224.

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