A Cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension). The condition for three general Cevians from the three vertices of a triangle to concur is known as Ceva's theorem.
Picking a Cevian point 
in the interior of a triangle 
and drawing Cevians from each vertex through 
to the opposite side produces a set of three intersecting
Cevians 
, 
, and 
with respect to that point. The triangle 
is known as the Cevian
triangle of 
with respect to 
,
and the circumcircle of 
is similarly known as the Cevian
circle.
If the trilinear coordinates of 
are 
,
then the trilinear coordinates of the points of intersection of the Cevians with
the opposite sides are given by 
, 
, and 
(Kimberling 1998, p. 185). Furthermore,
the lengths of the three Cevians are
 |  | ![(sqrt(bc[bc(beta^2+gamma^2)+(-a^2+b^2+c^2)betagamma]))/(bbeta+cgamma)](/images/equations/Cevian/Inline18.svg) |
(1)
|
 |  | ![(sqrt(ac[ac(alpha^2+gamma^2)+(a^2-b^2+c^2)alphagamma]))/(aalpha+cgamma)](/images/equations/Cevian/Inline21.svg) |
(2)
|
 |  | ![(sqrt(ab[ab(alpha^2+beta^2)+(a^2+b^2-c^2)alphabeta]))/(aalpha+bbeta).](/images/equations/Cevian/Inline24.svg) |
(3)
|
The ratios
 |
(4)
|
into which the Cevian point 
divides the Cevians have sum and ratio
 |  |  |
(5)
|
 |  |  |
(6)
|
which are respectively 
and 
(Ramler 1958; Honsberger 1995,
pp. 138-141).
