Stewart's Theorem

In geometry, Stewart's theorem yields a relation between the side lengths and a cevian length of a triangle. It can be proved from the law of cosines as well as by the famous Pythagorean theorem. Its name is in honor of the Scottish mathematician Matthew Stewart who published the theorem in 1746 when he was believed to be a candidate to replace Colin Maclaurin as Professor of Mathematics at the University of Edinburgh.

In $\triangle ABC$, point $D$ is a point on $BC$ and $AB=c, AC=b, BD=u, DC=v, AD=t.$ Stewart's theorem states that in this triangle, the following equation holds:

$$t^2=\frac{b^2u+c^2v}{u+v}-uv.$$

By the law of cosines, we have $$\begin{aligned} b^2&=v^2+t^2-2vt\cos \theta &\qquad (1)\\ c^2&=u^2+t^2+2ut\cos \theta. &\qquad (2) \end{aligned}$$ Now multiply (1) by $u$ and multiply (2) by $v$ to eliminate $\cos \theta$: $$\begin{aligned} b^2u&=uv^2+ut^2-2uvt\cos \theta &\qquad (3)\\ c^2v&=u^2v+vt^2+2uvt\cos \theta. &\qquad (4) \end{aligned}$$ Taking $(3)+(4)$ gives $$\begin{aligned} b^2u+c^2v&=uv(u+v)+t^2(u+v)\\ \Rightarrow t^2&=\frac{b^2u+c^2v}{u+v}-uv.\ _\square \end{aligned}$$

Stewart's theorem can sometimes be rewritten as $b^2u+c^2v=(u+v)(uv+t^2)$.

The proof below assumes $\angle B$ and $\angle C$ are both acute and $u < v$ as in the figure above. Then we have $$\begin{aligned} t^2 &= h^2 + x^2 \\ b^2 &= h^2 + (v-x)^2 \Rightarrow b^2u = h^2u + uv^2 - 2uvx + ux^2 \\ c^2 &= h^2 + (u+x)^2 \Rightarrow c^2v = h^2v + u^2v + 2uvx + vx^2, \end{aligned}$$ which implies $$\begin{aligned} b^2u + c^2v &= h^2u + h^2v + uv^2 + u^2v - 2uvx + 2uvx + ux^2 + vx^2 \\ &= (u + v)(h^2 + uv + x^2) \\ &= (u + v)(t^2 + uv) \\ &= a \cdot (t^2 + uv). \ _\square \end{aligned}$$

In the case where $\Delta ABC$ is isosceles (see figure above), Stewart's theorem has a more simplified form: $$\begin{aligned} a \cdot (t^2 + uv) &= b^2u + c^2v \\ &= b^2u + b^2v \\ &= b^2 (u + v) \\ &= ab^2 \\ \Rightarrow b^2 &= t^2 + uv. \end{aligned}$$ This theorem is quite useful in calculating the length of standard cevians like median, angle bisector, etc.