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 thatA common mnemonic used by students to memorize this equation 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 where the cevian is a median, 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.