Stewart's Theorem

Let a Cevian $A_1P$ be drawn on a Triangle $\Delta A_1A_2A_3$, and denote the lengths $m=\overline{A_2P}$ and $n=\overline{PA_3}$, with $a_1=m+n$. Then

$$m{a_2}^2+n{a_3}^2=(m+n)\overline{A_1P}\,^2+m\overline{PA_3}\,^2+n\overline{PA_2}\,^2.$$

This theorem is sometimes also called Apollonius' Theorem.

References

Altshiller-Court, N. ``Stewart's Theorem.'' §6B in College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, pp. 152-153, 1952.

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 6, 1967.