Lesson 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.

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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%5E2+m+%2B+c%5E2+n+=+a%28d%5E2+%2B+mn%29

b%5E2+m+%2B+c%5E2+n+=+a%28d%5E2+%2B+mn%29

This can be rewritten as follows:

man+%2B+dad+=+bmb+%2B+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?

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We can simply plug everything into the formula:

%283%29%287%29%284%29+%2B+7d%5E2+=+3%5E2%283%29+%2B+5%5E2%284%29

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