367] 
489 
367. 
ON A TRIANGLE IN-AND-CIRCUMSCRIBED TO A QUARTIC 
CURVE. 
[From the Philosophical Magazine, vol. xxx. (1865), pp. 340—342.] 
The quartic curve (a? 2 — a 2 ) 2 + (y 2 — ¿> 2 ) 2 = c 4 presents a simple example of a triangle 
in-and-circumscribed to a single curve, viz. such that each angle of the triangle 
is situate on, and each side touches, the curve. Assuming that the triangle is 
symmetrically situate in regard to the axis of y, viz. if it be the isosceles triangle 
ace, the sides whereof touch the curve in the points B, D, F respectively, then we 
must have a single relation between the constants a, h, c of the curve; or if (as may 
a 
c D e 
be done without loss of generality) we write a = 1, then there must be a single 
relation between h and c. The relation in question is most conveniently expressed by 
putting h and c equal to certain functions of a parameter </>, which is in fact = tan 2 0, 
if 6 be the angle at the base of the triangle; the equation of the curve is thus 
obtained in the form 
(# 2 -l) 2 + 
( . £+4£-iy 
V m<f> 2 +l)J 
= 1 
■ (0 2 -l) 4 . 
160 2 (</> 2 + l) 2 ’ 
c. v. 
62