Stewart's theorem
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 

Let , ,  be the lengths of the sides of a triangle. Let  be the length of a cevian to the side of length . If the cevian divides the side of length  into two segments of length  and , with  adjacent to  and  adjacent to , then Stewart's theorem states that

A common mnemonic used by students to memorize this equation (after rearranging the terms) is:

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

In the special case that the cevian is the median (that is, 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  be the angle between  and  and  the angle between  and . Then  is the supplement of , and so . Applying the law of cosines in the two small triangles using angles  and  produces

Multiplying the first equation by  and the third equation by  and adding them eliminates . One obtains

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 , ,  in terms of the altitude. The left and right hand sides of the equation then reduce algebraically to the same expression. History
According to , Stewart published the result in 1746 when he was a candidate to replace Colin Maclaurin as Professor of Mathematics at the University of Edinburgh. 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.  state that the result is used by Simson in 1748 and by Simpson in 1752, and its first appearance in Europe given by Lazare Carnot in 1803. See also 
 Mass point geometry

Notes

References

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. External links
 
 

Euclidean plane geometry
Theorems about triangles
Articles containing proofs