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In geometry, a cevian is a line segment which joins a vertex of a triangle to a point on the opposite side of the triangle. [1] [2] Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovanni Ceva, who proved a well-known theorem about cevians which also bears his name. [3]
Let a, b, c be the lengths of the sides of a triangle. Let d be the length of a cevian to the side of length a . If the cevian divides the side of length a into two segments of length m and n , with m adjacent to c and n adjacent to b , then Stewart's theorem states that b 2 m + c 2 n = a ( d 2 + m n ) . {\displaystyle b^{2}m+c^{2}n=a(d^{2}+mn).}
Ceva's theorem is a theorem of affine geometry, in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two line segments that are collinear). It is therefore true for triangles in any affine plane over any field.
The principle of calculation is that the foot of a cevian is the addition (defined above) of the two vertices (they are the endpoints of the side where the foot lie). For each cevian, the point of concurrency is the sum of the vertex and the foot. Each length ratio may then be calculated from the masses at the points. See Problem One for an ...
Every triangle inscribed in the circumcircle of the reference triangle ABC is congruent to exactly one circumcevian triangle. [2] The circumcevian triangle of P is similar to the pedal triangle of P. [2] The McCay cubic is the locus of point P such that the circumcevian triangle of P and ABC are orthologic. [3]
The triangle A'B'C' is the cevian triangle of Y. The triangle ABC and the cevian triangle A'B'C' are in perspective and let DEF be the axis of perspectivity of the two triangles. The line DEF is the trilinear polar of the point Y. The line DEF is the central line associated with the triangle center X.
The triangle A'B'C' is the cevian triangle of Y. The ABC and the cevian triangle A'B'C' are in perspective and let DEF be the axis of perspectivity of the two triangles. The line DEF is the trilinear polar of the point Y. DEF is the central line associated with the triangle center X.
In Euclidean geometry, a splitter is a line segment through one of the vertices of a triangle (that is, a cevian) that bisects the perimeter of the triangle. [1] [2] They are not to be confused with cleavers, which also bisect the perimeter but instead emanate from the midpoint of one of the triangle's sides.