Cevian
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)](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/Cevian/Inline18.gif) |
(1)
|
 |  | ![(sqrt(ac[ac(alpha^2+gamma^2)+(a^2-b^2+c^2)alphagamma]))/(aalpha+cgamma)](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/Cevian/Inline21.gif) |
(2)
|
 |  | ![(sqrt(ab[ab(alpha^2+beta^2)+(a^2+b^2-c^2)alphabeta]))/(aalpha+bbeta).](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/Cevian/Inline24.gif) |
(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).
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