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Stewart's Theorem

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StewartsTheorem

Let a Cevian PC be drawn on a triangle DeltaABC, and denote the lengths m=PA^_ and n=PB^_, with c=m+n. Then Stewart's theorem, also called Apollonius' theorem, states that

ma^2+nb^2=(m+n)PC^_^2+mn^2+nm^2.

In particular, if k is the fraction of the distance of P from vertex A to vertex B and k^'=1-k, then m=kc, n=k^'c, and

PC^_^2=a^2k-(c^2k-b^2)k^',

giving the above identity.

Bottema (1979) extended the formula to simplices in higher dimensions, and Bottema (1980-1981) explicitly considered the tetrahedron.

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References

Altshiller-Court, N. "Stewart's Theorem." §308 in College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, pp. 152-153, 1952.Bottema, O. "Eine Erweiterung der Stewartschen Formel." Elem. Math. 34, 138-140, 1979.Bottema, O. "De formule van Stewart voor een viervlak." Nieuw Tijdschr. Wisk., 68, 79-83, 1980-1981.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 6, 10, and 31, 1967.

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Stewart's Theorem

Cite this as:

Weisstein, Eric W. "Stewart's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StewartsTheorem.html

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