Routh's theorem states that if the sides , ,
and
of a nondegenerate triangle are divided at , ,
and ,
respectively, with ,
, and , where , , and are positive, the cevians, , and form a possibly degenerate central triangle
whose area is
for ,
2, 3, ..., the ratios
are 0, 1/7 (Steinhaus 1999, pp. 8-9), 4/13, 3/7, 16/31, 25/43, ... (OEIS A046162 and A046163).
The area of the triangle formed
by connecting the division points on each side is
(3)
The central-area formula gives Ceva's theorem when . With directed ratios and
interpreted as an oriented area,
the division-point formula gives Menelaus' theorem
when ,
provided the division points are finite (Mikusiński 1946).
Bottema, O. "On the Area of a Triangle in Barycentric Coordinates." Crux. Math.8, 228-231, 1982.Coxeter,
H. S. M. Introduction
to Geometry, 2nd ed. New York: Wiley, pp. 211-212, 1969.Dudeney,
H. E. Amusements
in Mathematics. New York: Dover, p. 27, 1970.Klamkin, M. S.
and Liu, A. "Three More Proofs of Routh's Theorem." Crux Math.7,
199-203, 1981.Mikusiński, J. G. "Sur quelques propriétés
du triangle." Ann. Univ. M. Curie-Sklodowska Sect. A1, 45-50,
1946.Sloane, N. J. A. Sequences A046162
and A046163 in "The On-Line Encyclopedia
of Integer Sequences."Steinhaus, H. Mathematical
Snapshots, 3rd ed. New York: Dover, 1999.
, 
,
and 
of a nondegenerate triangle 
are divided at 
, 
,
and 
,
respectively, with 
,
, and 
, where 
, 
, and 
are positive, the cevians 
, 
, and 
form a possibly degenerate central triangle
whose area is
is the area of the original triangle.
For 
,
,
2, 3, ..., the ratios 
are 0, 1/7 (Steinhaus 1999, pp. 8-9), 4/13, 3/7, 16/31, 25/43, ... (OEIS A046162 and A046163).
. With directed ratios and
interpreted as an oriented area,
the division-point formula gives Menelaus' theorem
when 
,
provided the division points are finite (Mikusiński 1946).
