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| In [[geometry]], '''Stewart's theorem''' yields a relation between a lengths of the sides of the [[triangle]] and the length of a [[cevian]] of the triangle. Its name is in honor of the Scottish mathematician [[Matthew Stewart (mathematician)|Matthew Stewart]] who published the theorem in 1746.<ref>M. Stewart ''Some General Theorems of Considerable Use in the Higher Parts of Mathematics'' (1746) "Proposition II"</ref>
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| == Theorem ==
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| Let <math>a</math>, <math>b</math>, and <math>c</math> be the lengths of the sides of a triangle. Let <math>d</math> be the length of a [[cevian]] to the side of length <math>a</math>. If the cevian divides <math>a</math> into two segments of length <math>m</math> and <math>n</math>, with <math>m</math> adjacent to <math>c</math> and <math>n</math> adjacent to <math>b</math>, then Stewart's theorem states that
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| :<math>b^2m + c^2n = a(d^2 + mn).\,</math>
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| [[Apollonius' theorem]] is the special case where d is the length of the [[Median (geometry)|Median]].
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| The theorem may be written somewhat more symmetrically using signed lengths of segments, in other words the length ''AB'' is taken to be positive or negative according to whether ''A'' is to the left or right of ''B'' in some fixed orientation of the line. In this formulation, the theorem states that if ''A'', ''B'', and ''C'' are collinear points, and ''P'' is any point, then<ref>Russell</ref> | |
| :<math>PA^2\cdot BC+PB^2\cdot CA-PC^2\cdot AB - BC\cdot CA\cdot AB =0.\,</math>
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| == Proof ==
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| [[File:Stewarts theorem.svg|right|300px|Diagram of Stewart's theorem]]
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| The theorem can be proved as an application of the [[law of cosines]]:<ref>Follows Hutton & Gregory or, more closely, PlanetMath.</ref>
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| Let ''θ'' be the angle between ''m'' and ''d'' and ''θ′'' the angle between ''n'' and ''d''. Then ''θ′'' is the supplement of ''θ'' and cos ''θ′'' = −cos ''θ''. The law of cosines for ''θ'' and ''θ′'' states
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| :<math>
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| \begin{align}
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| c^2 &= m^2 + d^2 - 2dm\cos\theta \\
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| b^2 &= n^2 + d^2 - 2dn\cos\theta' \\
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| &= n^2 + d^2 + 2dn\cos\theta.\, \end{align}
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| </math>
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| Multiply the first equation by ''n'', the second equation by ''m'', and add to eliminate cos ''θ'', obtaining
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| :<math>
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| \begin{align}
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| &b^2m + c^2n \\
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| &= nm^2 + n^2m + (m+n)d^2 \\
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| &= (m+n)(mn + d^2) \\
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| &= a(mn + d^2), \\
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| \end{align}
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| </math>
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| which is the required equation.
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| Alternatively, the theorem can be proved by drawing a perpendicular from the vertex of the triangle to the base and using the [[Pythagorean theorem]] to write the distances ''b'', ''c'', and ''d'' in terms of the altitude. The left and right hand sides of the equation then reduce algebraically to the same expression.<ref>This is a overview of the proof in Russell.</ref>
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| == See also ==
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| * [[Mass point geometry]]
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| ==References==
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| <references/>
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| 5. I.S Amarasinghe, Solutions to the Problem '''43.3''': Stewart's Theorem(''A '''New Proof''' for the Stewart's Theorem '''using Ptolemy's Theorem'''''), ''Mathematical Spectrum'', Vol '''43(03)''', pp.138 - 139, 2011.
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| 6. A. Ostermann, G. Wanner, Further Results in Euclidean Geometry: '''''Problem 14 of Exercises 4.11''''', '''Geometry by Its History'''(''Springer books''), '''pp. 112''', 2012.
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| ==Further reading==
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| *{{cite book |title=A Course of Mathematics|first1=C.|last1=Hutton|first2=O.|last2=Gregory |publisher=Longman, Orme & co.|year=1843|page=219|volume=II|url=http://books.google.com/books?id=9-4GAAAAYAAJ&pg=PA219#v=onepage}}
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| * {{cite book |title=Pure Geometry
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| |first=John Wellesley|last=Russell|publisher=Clarendon Press|year=1905
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| |chapter=Chapter 1 §3: Stewart's Theorem
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| |url=http://books.google.com/books?id=r3ILAAAAYAAJ|oclc= 5259132}}
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| ==External links==
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| * {{MathWorld|title=Stewart's Theorem|urlname=StewartsTheorem}}
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| * {{PlanetMath|title=Stewart's Theorem|urlname=StewartsTheorem}}
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| * {{PlanetMath|title=Proof of Stewart's Theorem|urlname=ProofOfStewartsTheorem}}
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| [[Category:Euclidean plane geometry]]
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| [[Category:Triangle geometry]]
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| [[Category:Articles containing proofs]]
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| [[Category:Theorems in plane geometry]]
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