Stewart's theorem

Stewart's theorem relates the side lengths of a triangle to the length of a cevian and the two segments it cuts on the opposite side.

Cevian: A segment from a triangle’s vertex to a point on the opposite side or its extension. Stewart’s theorem applies to the cevian and the two side segments it creates.
Median (geometry): A segment joining a triangle’s vertex to the midpoint of the opposite side. A median is the equal-segment case of the cevian in Stewart’s theorem.
Triangle: A polygon with three sides, three vertices, and three interior angles. Stewart’s theorem is a relation among lengths within a triangle.
Median: A line segment joining a polygon’s vertex to the midpoint of the opposite side or face. In triangle geometry, this general term narrows to the special cevian covered by Apollonius’s theorem.
Stewart's theorem formula: The identity d²a + b²m + c²n = a(mn + d²), where a = m + n and d is a cevian length. This is Stewart’s theorem written with the base split into lengths m and n.
Angle bisector theorem: A triangle theorem stating that an angle bisector divides the opposite side in the ratio of the adjacent sides. Its side division supplies the two base segments when Stewart’s theorem is applied to an angle bisector.
Side (geometry): A line segment forming one edge of a polygon. The theorem uses all three side lengths of the triangle.
Geometric proof: A proof that establishes a mathematical claim using geometric definitions, constructions, or theorems. Stewart’s theorem admits a proof by splitting the triangle and applying established length relations.
Apollonius's theorem: A triangle identity relating its side lengths to the median’s length. Setting the cevian’s two base segments equal reduces Stewart’s theorem to this median identity.
Triangle area: A measure equal to half a triangle’s base times its corresponding height. A known area and base can determine a triangle’s altitude, which may serve as a cevian.

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Ptolemy's inequalityRelated: Like Ptolemy's inequality, it converts geometric configuration constraints into length relations.