64 UNIVERSITY OF VIRGINIA PUBLICATIONS 



the points D, E, F into the boundary oi A B C ; and the points A^, B^, Cj 

 into the circumference of the circnmcircle oi AB C. The equation of the 

 circumcircle in x, y, z can be written 



cTi, o-j, 0-3 being linear, and ambiguities of sign allowed. 



16. There are many interesting theorems closely associated with the 

 problem we have been discussing; we refer to only three of them. 



(1). To find the point in a plane at which sides of a given triangle 

 subtend given angles. We have only to take a, /?, y proportional to the 

 sines of the given angles, having regard for the algebraic signs of the angles. 



(3). Theorems relating to the groups of triangles each one of which 

 is similar to a given triangle, in — and circumscribed about a given 

 triangle. Special reference may be made to three papers by Mr. John 

 Griffiths, London Math. Soc. XXIV, pp. 131, 181, 369. Mr. Griffiths calls 

 the ratios (31) the isogonal coordinates of P. 



(3). The maximum and minimum triangles of given form circum- 

 scribed and inscribed about a given triangle ABC. The former has its 

 sides perpendicular to r^, r^, r^ at P, and if a, p, y are the sides of A'B'C 

 similar to the given form, and M the area of the maximum triangle A^^B^C^ 

 then 



a,/a = b^/p = c^/y = lc = 2M/s = s/2B, (38) 



determines this triangle. 



(4). The reciprocal or dual relations existing between the two tri- 

 angles ABC and A'B'C in all the formulte we have deduced are striking 

 and admit interesting relations. 



17. The following algebraic determination of s and P is interesting. 

 Starting with the necessary conditions for an ordinary maximum 01 

 minimum 



p y s 



= r.^- ^ c- y -\- b'^ z — a-, 



= x{x y c- -\- X z &-) — X- cr, 



^ x{(x — y z a-) — X- cr, 



= {x y -\- X z)(j — X y z a-. (39) 



