242 
ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. [514 
The reduction is due first and secondly to triangles wherein the angle a coincides 
with an angle c or e, and thirdly to triangles wherein the angles a, c, e all coincide. 
1°. Take for the side cDe a double tangent of the curve B = D = F, this meets 
the curve a= c = e in x points, and selecting any one of them for e and any other for c, 
Fig. 5. 
we have from the last-mentioned point Y-2 tangents to the curve B = D = F; and 
in respect of each of these a position of a coincident with c. The reduction on this 
account is 2tx(x — 1)(Y— 2); but since we may in the figure interchange c and e, 
B and F, we have the same number belonging to the coincidence of the angles a, e, 
or together the reduction is = 4>tx(x — 1)(Y — 2). 
Fig. 6. 
But instead of a double tangent we may have cDe a stationary tangent; we have 
thus reductions x{x — \){Y — 2) and '3tx(x— 1)(F — 2), together 6ix (x — 1) (F — 2) ; 
and for the double and stationary tangents together we have 
(4t + 6i) x (x — 1) ( F — 2), 
= 2 {F(F— 1) —y) x(x— 1) (F—2), 
that is 
= 2x(x — 1) F(F— 1)(F— 2) - 2x (x— 1 )y(Y— 2). 
2°. The side cBe may be taken to be a tangent to the curve B = D = F at any 
one of its intersections with the curve a — c==e. Taking then the point e at the 
intersection in question, and the point c at any other of the intersections of the 
tangent with the curve a = c= e, and from c drawing any other tangent to the curve 
B = D = F, there is in respect of each of these tangents a position of a at c; and