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XV. On the Problem of the In-and-Circumscribed Triangle. By A. Cayley, F.It.S. 
Received December 30, 1870, — Read February 9, 1871. 
The problem of the In-and-Circumscribecl Tiiangle is a particular case of that of the 
In-and-Circumscribed Polygon : the last-mentioned problem may be thus stated — to find 
a polygon such that the angles are situate in and the sides touch a given curve or curves. 
And we may in the first instance inquire as to the number of such polygons. In the 
case where the curves containing the angles and touched by the sides respectively are 
all of them distinct curves, the number of polygons is obtained very easily and has a 
simple expression : it is equal to twice the product of the orders of the curves containing 
the several angles respectively into the product of the classes of the curves touched by 
the several sides respectively ; or, say, it is equal to twice the product of the orders of 
the angle-curves into the product of the classes of the side-curves. But when several of 
the curves become one and the same curve, and in particular when the angles are all of 
them situate in and the sides all touch one and the same curve, it is a much more diffi- 
cult problem to find the number of polygons. The solution of this problem when the 
polygon is a triangle, and for all the different relations of identity between the different 
curves, is the object of the present memoir, which is accordingly entitled “ On the Pro- 
blem of the In-and-Circumscribed Triangle the methods and principles, however, are 
applicable to the case of a polygon of any number of sides, the method chiefly made use 
of being that furnished by the theory of correspondence, as will be explained. The 
results (for the triangle) are given in the following Table ; for the explanation of which 
I remark that the triangle is taken to be aBcDeF ; viz. a, c, e are the angles, B, D, F 
the sides ; that is, B, D, F are the sides ac , ce , ea respectively, and a , c, e are the angles 
FB, BD, DF respectively. And I use the same letters a, c, e, B, D, F to denote the 
curves containing the angles and touched by the sides respectively ; viz. the angle a is 
situate in the curve a, the side B touches the curve B, and so for the other angles and 
sides respectively. An equation such as a—c or a— B denotes that the curves «, c or, 
as the case may be, the curves «, B are one and the same curve : it is in general con- 
venient to use a new letter for denoting these identical curves ; viz. I write, for instance, 
a—c—x or a=I>=x, to denote that the curves «, c or, as the case may be, the curves 
a, B are one and the same curve x ; the new letters thus introduced are x, y , z, there 
being in regard to them no distinction of small letters and capitals. The expression 
“no identities” denotes that the curves are all distinct. But I use also the letters a, c, 
e, b, d,f, x, y , z, and A, C, E, B, 19, F, X, Y, Z quantitatively, to denote the orders and 
classes of the curves a , c, e, B, D, F, x, y, z respectively ; thus, in the Table, for the 
case 1 “ no identities” the number of triangles is given as =2«ccBDF, which agrees witli 
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