PRINCIPIA · THEOREM

Stewart's theorem

Dependencies: Pythagorean theorem.

Statement

Let DD be a point on side BCBC of △ABC\triangle ABC (i.e. a cevian ADAD). Write

m=∣BD∣,n=∣DC∣,d=∣AD∣,m = |BD|,\qquad n = |DC|,\qquad d = |AD|,

and, by the standard convention, the three sides

a=∣BC∣=m+n,b=∣CA∣,c=∣AB∣.a = |BC| = m + n,\qquad b = |CA|,\qquad c = |AB|.

Then we have the Stewart identity

b2 m  +  c2 n  =  a (d2+m n).b^{2}\,m \;+\; c^{2}\,n \;=\; a\,\bigl(d^{2} + m\,n\bigr).

\triangle ABC with cevian AD cuts BC into BD = m and DC = n; the formula b^{2}m + c^{2}n = a(d^{2} + mn).

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