Lesson Stewart's Theorem

Source code of 'Stewart's Theorem'

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Stewart's theorem yields a relation between the lengths of the sides and the length of a cevian in a triangle. A cevian is a line that passes through a triangle's vertex and a point on the side opposite that vertex.

*[illustration stewart_s_theorem.gif].

In the following illustration, a, b, c are the sides of the triangles, m, n is the length of the segments split by the cevian, and d (not labeled) is the length of the cevian.

We have the following relationship:

{{{b^2 m + c^2 n = a(d^2 + mn)}}}

This can be rewritten as follows:

{{{man + dad = bmb + cnc}}}

Remember, "a man and a dad put a bomb in the sink."

Let us begin with an example. Suppose we have a triangle with sides AC = 3, AB = 5, and BC = 7. The cevian splits side BC into segments of length 3 and 4. What is the length of the cevian?

*[illustration stewart_s_theorem_v2.png].

We can simply plug everything into the formula:

{{{(3)(7)(4) + 7d^2 = 3^2(3) + 5^2(4)}}}

{{{84 + 7d^2 = 127}}}

{{{7d^2 = 43}}}

{{{d = sqrt(43)/sqrt(7) = 2.478}}}