Stewart's theorem

Template:Short description In geometry, Stewart's theorem yields a relation between the lengths of the sides and the length of a cevian in a triangle. Its name is in honour of the Scottish mathematician Matthew Stewart, who published the theorem in 1746.

Statement

Diagram of Stewart's theorem

Let Template:Mvar, Template:Mvar, Template:Mvar be the lengths of the sides of a triangle. Let Template:Mvar be the length of a cevian to the side of length Template:Mvar. If the cevian divides the side of length Template:Mvar into two segments of length Template:Mvar and Template:Mvar, with Template:Mvar adjacent to Template:Mvar and Template:Mvar adjacent to Template:Mvar, then Stewart's theorem states that $$b^{2}m+c^{2}n=a(d^2+mn).$$

A common mnemonic used by students to memorize this equation (after rearranging the terms) is: $$\underset{\text{A }man\text{ and his }dad}{man+dad}=\underset{\text{put a }bomb\text{ in the }sink.}{bmb+cnc}$$

The theorem may be written more symmetrically using signed lengths of segments. That is, take the length Template:Mvar to be positive or negative according to whether Template:Mvar is to the left or right of Template:Mvar in some fixed orientation of the line. In this formulation, the theorem states that if Template:Mvar are collinear points, and Template:Mvar is any point, then

$(\stackrel{2}{\stackrel{‾}{PA}}\cdot \overline{BC})+(\stackrel{2}{\stackrel{‾}{PB}}\cdot \overline{CA})+(\stackrel{2}{\stackrel{‾}{PC}}\cdot \overline{AB})+(\overline{AB}\cdot \overline{BC}\cdot \overline{CA})=0.$

In the special case where the cevian is a median (meaning it divides the opposite side into two segments of equal length), the result is known as Apollonius' theorem.

Proof

The theorem can be proved as an application of the law of cosines.

Let Template:Mvar be the angle between Template:Mvar and Template:Mvar and Template:Mvar the angle between Template:Mvar and Template:Mvar. Then Template:Mvar is the supplement of Template:Mvar, and so Template:Math. Applying the law of cosines in the two small triangles using angles Template:Mvar and Template:Mvar produces $$\begin{array}{cc}{c}^2& =m^2+d^2-2dm\cos \theta ,\\ b^2& =n^2+d^2-2dn\cos {\theta }^{\prime }\\ & =n^2+d^2+2dn\cos \theta .\end{array}$$

Multiplying the first equation by Template:Mvar and the third equation by Template:Mvar and adding them eliminates Template:Math. One obtains $$\begin{array}{cc}{b}^{2}m+c^{2}n& =n{m}^2+n^{2}m+(m+n)d^2\\ & =(m+n)(mn+d^2)\\ & =a(mn+d^2),\\ \end{array}$$ which is the required equation.

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 Template:Mvar, Template:Mvar, Template:Mvar in terms of the altitude. The left and right hand sides of the equation then reduce algebraically to the same expression.

History

According to Template:Harvtxt, Stewart published the result in 1746 when he was a candidate to replace Colin Maclaurin as Professor of Mathematics at the University of Edinburgh. Template:Harvtxt state that the result was probably known to Archimedes around 300 B.C.E. They go on to say (mistakenly) that the first known proof was provided by R. Simson in 1751. Template:Harvtxt state that the result is used by Simson in 1748 and by Simpson in 1752, and its first appearance in EuropeTemplate:Clarification needed given by Lazare Carnot in 1803.

See also

• Mass point geometry

References

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Further reading

• 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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External links

• Template:MathWorld
• Template:PlanetMath