Stewart's theorem

Statement

Let $ABC$ be a triangle in the Euclidean plane, and let $D$ be a point on the segment $BC$ . Write $a = BC$ , $b = CA$ , $c = AB$ , $m = BD$ , $n = DC$ and $d = AD$ for the distances. Then $b^2 m + c^2 n = a \left( d^2 + m n \right).$

Remarks

If $D$ lies strictly between $B$ and $C$ , the angles $\angle ADB$ and $\angle ADC$ add up to $\pi$ , so their cosines are opposite. Apply the law of cosines to the triangles $ABD$ and $ADC$ at the vertex $D$ . Multiply the first equation by $n$ , the second by $m$ , and add: the cosine terms cancel, and $m + n = a$ gives the formula. If $D = B$ or $D = C$ , the formula is direct.

Example: if $D$ is the midpoint of $BC$ , then $m = n = a/2$ , and the formula becomes $d^2 = \frac{b^2 + c^2}{2} - \frac{a^2}{4}$ . This is Apollonius's theorem on the length of a median.

For a point $D$ on the line $BC$ outside the segment, the formula holds with signed lengths $m$ and $n$ .

Going deeper

For a student

Stewart's theorem is the general form of several familiar length formulas. With the midpoint it gives the length of a median. With the foot of an angle bisector, located by the angle bisector theorem, it gives the length of the bisector.

Reading path

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Euclidean distance concept
Triangle concept
Stewart's theorem statement

Tag	`HXSK0BNBNE`
URI	`m2://HXSK0BNBNE`
Kind	statement, theorem
Writing	W0 stub
MSC	51M04
Status	This copy of the site has no build data. Run `m2 status HXSK0BNBNE`.
About	Triangle
Uses	Triangle, Euclidean distance
Same as	Wikidata Q739403 Wikipedia Stewart's theorem