468 ON A SYSTEM OF EQUATIONS CONNECTED WITH MALFATTl’s PROBLEM, [79 
where r 2 ——^—, and the solution is 
a + b + c 
x = ^-{a + b + c — r + *J{r 2 + a 2 ) — VO" 2 + & 2 ) — VO" 2 + c 2 )}, 
*uCl 
y = ^{a + 6 + c — r + VO" 2 + a 2 ) + VO" 2 + 6 2 ) — VO" 2 + c 2 )}, 
z =^{a + b + c — r — VO" 2 + a 2 ) — VO" 2 + 6 2 ) + VO" 2 + c 2 )}, 
V(yz) = i {r - a + V(^ 2 + a 2 )}, 
V(zx)= 1 {r - b + VO" 2 + 6 2 )}, 
V(xy) = i {r - C + V(^ 2 + C 2 )}, 
a system of formulae which contain the solution of the problem “In a given triangle 
to inscribe three circles such that each circle touches the remaining two circles and 
also two sides of the triangle.” In fact, if r denote the radius of the inscribed circle, 
and a, b, c the distances of the angles of the triangle from the points where the sides 
are touched by the inscribed circle (quantities which it is well known satisfy the con 
dition r 2 = —— ), also if x, y, z denote the radii of the required circles, there is no 
difficulty whatever in obtaining for the determination of x, y, z, the above system of 
equations. The problem in question was first proposed and solved by an Italian 
geometer named Malfatti, and has been called after him Malfatti’s problem. His solu 
tion, dated 1803, and published in the 10th volume of the Transactions of the Italian 
Academy of Sciences, appears to have consisted in showing that the values first found 
for the radii of the three circles satisfy the equations given above, without any indi 
cation of the process of obtaining the expressions for these radii. Further information 
as to the history of the problem may be found in the memoir “ Das Malfattische 
Problem neu gelost von C. Adams,” Winterthur, 1846. 
In connexion with the preceding investigations may be considered the problem of 
determining l and m from the equations 
B (l + O') 2 — 2 H (l + 0) m + (A + 1) m 2 = 0, 
A (m + 6) 2 — 2H (ra + 6)1 + (B + 1) l 2 = 0 ; 
which express that the function 
6 2 U + (lx + my) 2 , ( U = Ax 2 + 2Hxy + By 2 ), 
has for one of its factors a factor of U + x 2 , and for the other of its factors a factor 
of U + y 2 . There is no difficulty in solving these equations ; and if we write 
K — AB — H 2 , cti = V(— K — B), = V(— K — A).