# Who was Burlet, the one from Burlet's theorem?

The Burlet's theorem is a result in Euclidean geometry, which can be formulated as follows:

Theorem. Consider triangle $$ABC$$ with $$\angle{C}=\gamma$$. Let $$P$$ be the point where the incircle touches side $$AB$$, and denote the lengths $$AP$$ and $$BP$$ as $$m$$ and $$n$$, respectively. Then

$$\Delta=mn\cot{\frac{\gamma}{2}},$$

where $$\Delta$$ denotes the area of $$ABC$$. Note that when $$\gamma=\frac{\pi}{2}$$, then $$\Delta=mn$$ (which is more well-known).

A proof and an extension to cyclic quadrilateral can be found here.

So, who was Burlet really? It seems that in Latin America, this theorem is known by this name, but no one really knows who he was. I have found some references to Oscar Burlet here, here and here. Could it be the same person?

• Yes, I think this is related. Commented Jan 5 at 6:36
• @njuffa: Thank you very much. That reference will be useful for me. If you'd like, you can turn your comments into an answer, and I'll accept it as satisfactory. Commented Jan 5 at 6:52
• @njuffa maybe you should post this as an answer Commented Jan 5 at 12:18

I am unable to find any references to a "theorem of Burlet" in a print publication regardless of language, e.g. Satz von Burlet, théorème de Burlet. All online resources referencing this term are in Portuguese and are recent. For example, the entry in the Portuguese Wikipedia linked in the question dates to 2023.

I performed an internet search for "Burlet" in conjuction with "triangle" and "inscribed circle" in multiple languages which yielded a relevant question by an A. Burlet from Dublin:

Nouvelles annales de mathématiques, Vol. 15, 1856, p. 290:

Questions. [...] 336. Un triangle rectangle est équivalent au rectangle des deux segments faits sur l'hypoténuse par le point de contact du cercle inscrit. (A. BURLET de Dublin)

Translation: "A right triangle is equivalent to the rectangle of the two segments made on the hypotenuse by the point of contact of the inscribed circle." A solution to the problem posed by Burlet appeared in the same publication the following year:

P. H. Rochette, S.J., "Solution de la Question 336", Nouvelles annales de mathématiques, Vol. 16, 1857, pp. 43-44.

I am unable to find further information on A. Burlet in other publications of the time. Burlet may have been a teacher or amateur mathematician. Presumably Dublin refers to the Irish city but we cannot be entirely sure.

• Anecdotical remark : "S.J." after the name "Rochette" means "Société de Jésus" ; Mr Rochette was a jesuit priest (not the first mathematician belonging to this congregation). Commented Jan 5 at 23:54
• 1) I think that somebody signing S.J. has been ordinated as a priest. But I must check it 2) I don't believe there is a direct connection between Rochette (the Jesuit) and Burlet. Commented Jan 6 at 6:43
• I wonder if "Burlet" may be a mis-spelling of a name like "Burleigh" or "Burley"? I have searched for both possibilities without success. An Arthur George Burleigh graduated B.A. from the University of Dublin in 1866, but (1) this seems a bit late and (2) I have been unable to establish the subject of his degree. Commented Jan 8 at 19:53