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
(1)
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(2)
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(3)
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The ratios
(4)
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into which the Cevian point divides the Cevians have sum and ratio
(5)
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(6)
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which are respectively and (Ramler 1958; Honsberger 1995, pp. 138-141).