128] 
IN-AND-CIRCUMSCRIBED POLYGON. 
143 
by concentric circles. Thus, in the case of a triangle, if ABC be the triangle, and 
a, /3, 7 be the points of contact of the sides with the inscribed conic, then the tangents 
to the circumscribed conic at A, B, C meet the opposite sides BC, CA, AB in points 
lying in the chord of contact, the lines A a, B/3, Cy meet in the pole of contact, 
and so on. 
In the case of a quadrangle, if ACEG be the quadrangle, and b, d, f h the 
points of contact with the inscribed conic, then the tangents to the circumscribed 
conic at the pair of opposite angles A, E and the corresponding diagonal CG, and 
in like manner the tangents at the pair of opposite angles C, G and the corresponding 
diagonal AE, meet in the chord of contact. Again, the pairs of opposite sides AC, 
EG, and the line dh joining the points of contact of the other two sides with the 
inscribed conic, and the pairs of opposite sides AG, CE, and the line bf joining the 
points of contact of the other two sides with the inscribed conic, meet in the chord 
of contact. The diagonals AE, CG, and the lines bf, dh through the points of 
contact of pairs of opposite sides with the inscribed conic, meet in the pole of 
contact, &c. 
The beautiful systems of ‘ focal relations ’ for regular polygons (in particular for 
the pentagon and the hexagon), given in Sir W. R. Hamilton’s Lectures on Quaternions, 
[Dublin, 1853] Nos. 379—393, belong, it is clear, to polygons which are inscribed in and 
circumscribed about two conics having double contact with each other. In fact, the focus 
of a conic is a point such that the lines joining such point with the circular points at 
infinity (i.e. the points in which a circle is intersected by the line infinity) are tangents 
to the conic. In the case of two concentric circles, these are to be considered as 
touching in the circular points at infinity; and consequently, when the concentric 
circles are replaced by two conics having double contact, the circular points at infinity 
are replaced by the points of contact of the two conics.