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Cevian

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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.

Cevians

Picking a Cevian point P in the interior of a triangle DeltaABC and drawing Cevians from each vertex through P to the opposite side produces a set of three intersecting Cevians AA^', BB^', and CC^' with respect to that point. The triangle DeltaA^'B^'C^' is known as the Cevian triangle of DeltaABC with respect to P, and the circumcircle of DeltaA^'B^'C^' is similarly known as the Cevian circle.

If the trilinear coordinates of P are (alpha,beta,gamma), then the trilinear coordinates of the points of intersection of the Cevians with the opposite sides are given by (0,beta,gamma), (alpha,0,gamma), and (alpha,beta,0) (Kimberling 1998, p. 185). Furthermore, the lengths of the three Cevians are

AA^'^_=(sqrt(bc[bc(beta^2+gamma^2)+(-a^2+b^2+c^2)betagamma]))/(bbeta+cgamma)
(1)
BB^'^_=(sqrt(ac[ac(alpha^2+gamma^2)+(a^2-b^2+c^2)alphagamma]))/(aalpha+cgamma)
(2)
CC^'^_=(sqrt(ab[ab(alpha^2+beta^2)+(a^2+b^2-c^2)alphabeta]))/(aalpha+bbeta).
(3)

The ratios

r_A,r_B,r_C=(AP)/(PA^'),(BP)/(PB^'),(CP)/(PC^')
(4)

into which the Cevian point P divides the Cevians have sum and ratio

r_A+r_B+r_C=(bbeta+cgamma)/(aalpha)+(aalpha+cgamma)/(bbeta)+(aalpha+bbeta)/(cgamma)
(5)
r_Ar_Br_C=((aalpha+bbeta)(aalpha+cgamma)(bbeta+cgamma))/(abcalphabetagamma),
(6)

which are respectively >=6 and >=8 (Ramler 1958; Honsberger 1995, pp. 138-141).

In a triangle ABC, suppose AB, CA, and CB have finite slopes r, t, and s, respectively. If M=(1-u)A+uB, the slope m of the Cevian CM satisfies (Mhanna 2025, p. 1)

1/(m-r)=(1-u)/(t-r)+u/(s-r),
(7)

where the left-hand side is interpreted as zero when CM is vertical. When AB is horizontal, the formula linearly interpolates the cotangents of the inclinations of CA and CB to give that of CM.

See also

Angle Bisector, Ceva's Theorem, Cevian Circle, Cevian Point, Cevian Simplex, Cevian Triangle, Pedal-Cevian Point, Routh's Theorem, Splitter, Triangle Median

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References

Honsberger, R. "On Cevians." Ch. 12 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 13 and 137-146, 1995.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Mhanna, A. "A Cevian Slope Formula with Applications." 2025. https://www.researchgate.net/publication/417629658_A_CEVIAN_SLOPE_FORMULA_WITH_APPLICATIONS.Ramler, O. J. Solved by C. W. Trigg. "Problem E1043." Amer. Math. Monthly 65, 421, 1958.Thébault, V. "On the Cevians of a Triangle." Amer. Math. Monthly 60, 167-173, 1953.

Referenced on Wolfram|Alpha

Cevian

Cite this as:

Weisstein, Eric W. "Cevian." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Cevian.html

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