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67 
175. 
ON THE PORISM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. 
•is, 1705, 
marginal 
intended 
no con- 
[From the Quarterly Mathematical Journal, vol. I. (1857), pp. 844—354.] 
The porism of the in-and-circumscribed triangle in its most general form relates 
to a triangle the angles of which lie in fixed curves, and the sides of which touch 
fixed curves; but at present I consider only the case in which the angles lie in one 
and the same fixed curve, which for greater simplicity I assume to be a conic. We 
have therefore a triangle ABC, the angles of which lie in a fixed conic ©, and the 
sides of which touch the fixed curves 31, 33, 6; the points of contact may be repre 
sented by a, ¡3, y. And if we consider the conic © and the curves 31, 33 as given, 
the curve (£ will be the envelope of the side AB; to construct this side we have 
only to take at pleasure a point C on the curve © and to draw through this point 
tangents to the curve 33, 31 respectively meeting the conic © in the points A and B ; 
the line joining these points is the required side AB. I may notice that in the 
case supposed of the curve © being a conic, the lines A a, B/3, Cy meet in a point; 
which gives at once a construction for y, the point of contact of AB with the curve C. 
For the sake however of exhibiting the reasoning in a form which may be modified 
so as to be applicable to a curve © of any order, instead of the conic ©, I shall 
dispense with the employment of the property just mentioned, which is peculiar to 
the case of the conic. 
Suppose for a moment (figs. 1 and 1 bis) that the curves 31, 33 are points, and let 
the line through 31, 33 meet the conic © in the points M, N. If we take the point 
N for the angle C of the triangle, the points A, B will each of them coincide with 
M, and the side AB will be the tangent at M to the conic © : call this tangent T. 
Consider next a point C in the neighbourhood of N, we shall have two points A, B 
in the neighbourhood of M, and the point in which AB intersects T will be the 
point of contact y of T with the curve (£. To find this point, suppose that M31 = a, 
i/33 = 6, N11 = a', N£ = b' and let the distance of C from N be ds\ the distances 
parallel to T of A, B from the line MN will be proportional to y, ds, ~ds, and 
9—2