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Diagram of Stewart's theorem. 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 + = (+).
The length of a cevian can be determined by Stewart's theorem: in the diagram, the cevian length d is given by the formula + = (+). Less commonly, this is also represented (with some rearrangement) by the following mnemonic:
Routh's theorem - Many problems involving triangles with cevians will ask for areas, and mass points does not provide a method for calculating areas. However, Routh's theorem , which goes hand in hand with mass points, uses ratios of lengths to calculate the ratio of areas between a triangle and a triangle formed by three cevians.
Steinitz theorem (graph theory) Stewart's theorem (plane geometry) Supporting hyperplane theorem (convex geometry) Sylvester–Gallai theorem (plane geometry) Szemerédi–Trotter theorem (combinatorics) Tverberg's theorem (discrete geometry) Vitali covering theorem (measure theory) Wallace–Bolyai–Gerwien theorem (discrete geometry)
Proof of Apollonius's theorem. The theorem can be proved as a special case of Stewart's theorem, or can be proved using vectors (see parallelogram law). The following is an independent proof using the law of cosines. [1] Let the triangle have sides ,, with a median drawn to side .
He was born in the manse at Rothesay, on the Isle of Bute, on 15 January 1717, [3] the son of Rev Dugald Stewart, the local minister, and his wife, Janet Bannantyne. [4]He was educated at Rothesay Grammar School, then entered the University of Glasgow in 1734, where he studied under the philosopher Francis Hutcheson and the mathematician Robert Simson, the latter from whom he studied ancient ...
This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just the particular case of the above formula where a ( x ) = a ∈ R {\displaystyle a(x)=a\in \mathbb {R} } is constant, b ( x ) = x , {\displaystyle b(x)=x,} and f ( x , t ...
Routh's theorem. In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians.The theorem states that if in triangle points , , and lie on segments , , and , then writing =, =, and =, the signed area of the triangle formed by the cevians , , and is