ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED 
TRIANGLE. 
[From the Philosophical Transactions of the Royal Society of London, vol. clxi. (for the 
year 1871), pp. 369—412. Received December 30, 1870,—Read February 9, 1871.] 
The problem of the In-and-Circumscribed Triangle 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 difficult 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 Problem of the In-and-Circum 
scribed 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