514] ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE.
243
the reduction on this account is = xy (x — 1) (F — 1). But interchanging in the figure
the letters c, e, B, F, there is an equal reduction belonging to the coincidence of a, e\
and the whole reduction in this manner is = 2x (x — i)y(r-i).
Fig. 7.
3°. If the side cDe intersects the curve a= c = e in two coincident points, then
taking these in either order for the points c, e, and from the two points respectively
drawing two other tangents to the curve D = B = F, we have a triangle wherein the
angles a, c, e all coincide. The side cDe may be a proper tangent to the curve
ci = c — e, or it may pass through a node or a cusp of this curve, viz. it is either a
common tangent of the curves B = JD = F and a=c = e (as in the figure, except that for
greater distinctness the points c and e are there drawn nearly instead of actually
coincident), or it may be a tangent to the curve B = D= F from a node or a cusp of
the curve a = c = e; we have thus the numbers
Common tangent XY (F — 1)(F — 2),
Tangent from node 2SF(F — 1) (F — 2),
Tangent from cusp 2kY(F— 1)(F — 2);
but (as we are counting intersections with the curve a = c = e) the second of these, as
being at a node of this curve, is to be taken 2 times; and the third, as being at a
cusp, 3 times; and the three together are thus
(X + 4S + 6* ) Y(Y — 1) (F— 2),
= {2x(x-l)-X} F(F—1)(F—2).
The reductions 1°, 2°, 3° altogether are
which is
2x(x-l) F(F —1)(F—2)
— 2x (x — 1) y(Y — 2)
+ 2x(x-l)y(Y - 1)
+ 2x(x-l) F(F— 1) (F— 2)
-IF(F-l)(F-2),
= 4x(x — 1) F (F— 1)(F —2)
+ 2x(x — 1) y
-IF(F-1)(F—2);
31—2