514] ON THE PROBLEM OF THE IN-AND-CIRCUMSCRIBED TRIANGLE. 241 
where the last term aDtc arises from the tangents cBa and eFa, each coinciding with 
a cuspidal tangent, as shown in the figure. 
Fig. 4. 
The second-mode reduction is 
= aD . A (x — 2) (x — 3), 
so that the two reductions together are 
= aD {(A - 2) A + (A - 4) 28 + (A - 3) 3« + * + A (x - 2) (x - 3)}, 
viz. this is 
= aD {(A — 2) A -f- (A — 4) (23 + 3k) + 4k 4- A (cc — 2) (x — 3)} ; 
or substituting for 23 + 3k and k the values x 2 — x — X and — 3A + £ respectively, and 
reducing, it is 
aD {A (2x 2 — 6x — 4) — 4x 2 + 4x + }. 
Hence subtracting from % + %', written in the form 
aD {A 2 (2x 2 — Qx + 4) + A (— 4x 2 + 12x — 8)}, 
the result is 
= aD (A 2 {2x 2 — Qx + 4) + A (— 6x 2 + 18ic — 4) + 4a; 2 — 4x — 4f}. 
On account of the symmetry we must divide by 2. 
Case 43. a — c = e = x, B — D — F =y. 
Suppose for a moment that the angle a is a free point; the locus of a is a curve 
the order of which is obtained from Case 28, by writing c = e = x, B — D = F — y \ the 
locus in question meets a curve order a in {2F(F— 1) (F— 2) + y) x (x — l)a points; 
wherefore the order of the locus is 
= {2F(F— 1) (F— 2) + y\ x (x — 1), 
and this locus meets the curve a = c = e = x in a number of points 
= {2F(F — l)(Y—2) + y}x 2 {x— 1), 
viz. this is the number of positions of the angle a; but several of these belong to 
special forms of the triangle aBcDeF, giving heterotypic solutions, which are to be 
rejected ; the required number is thus 
{2F(F— 1) (F —2) + y) x 2 {x — — Red. 
C. VIII. 
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