267] 
ON THE PORISM OF THE IN-AND-CIRCUMSCRIBED POLYGON. 
307 
This is verified very simply in the case of the quadrangle. Taking for the two 
points of intersection the circular points at infinity, the line joining them is the line 
infinity, and its pole (with respect to the inscribed circle) is the centre of this circle; 
the relation therefore is that the centre of the inscribed circle lies on the circumscribed 
circle. But when this is the case, it is easy to see that the pole (w r ith respect to 
the inscribed circle) of the radical axis, lies also on the circumscribed circle ; this pole 
and the centre of the inscribed circle are in fact the extremities of a diameter of the 
circumscribed circle. The condition thus obtained is R 2 — a 2 = 0 (which is M. Mention’s 
condition ¿=0). We have next to find the analytical relation when the pole (with 
respect to the inscribed circle), of the line joining one of the actual points of inter 
section with a circular point at infinity is a point on the circumscribed circle. This 
I effect as follows:—taking z = 0 as the equation of line infinity, if the origin be 
taken on the middle point of the radical axis, and if x = 0 be the radical axis, then 
the equations of the two circles may be taken to be 
Inscribed circle, x 2 + y 2 — 21 xz — Vz 2 = 0, 
Circumscribed circle, x 2 + y 2 — ILxz — Vz 2 = 0, 
a circular point at infinity is 
x : y : z = 1 : i : 0, (i = V — 1), 
an actual point of intersection is 
x : y : z = 0 : W : 1. 
The line joining these is 
xi —y + z VV = 0, 
its pole, with respect to the inscribed circle, is 
x : y : z = — tVV : VV — il : 1; 
and if this be a point of the circumscribed circle 
— V + (V — HI — l 2 ) + 2 Li \/V-V = 0, 
that is 
2 (Z-o^Vv = ^ + v, 
or 
(l 2 + V) 2 + 4,(L-l) 2 V = 0, 
which is the required relation: but to express it in terms of the ordinary data R, r, a, 
the equations of the circles, putting therein z= 1, become 
(x — l ) 2 + y 2 = V +1 2 , 
(x-L) 2 + y 2 = V +L 2 , 
39—2