Stewart's Theorem

Author

Eric W. Weisstein
October 5, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/StewartsTheorem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/StewartsTheorem.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Derivation

PC=FullSimplify[Distance[{0,0,1},{(1-k)b,k a,0}]^2]

FullSimplify[m a^2+n b^2/.{m->k c,n->(1-k)c}]

FullSimplify[(m+n)PC^2+m n^2+n m^2/.{m->k c,n->(1-k)c}]

Figure

With[{
t=Triangle[{{0,0},{1,0},{.3,.5}}],
k=.4
},
Show[Graphics[{
t,
Line[Coordinates/@(l=Cartesian[#[t]]&/@{{0,0,1},{(1-k)b,k a,0}}/.Thread[{a,b,c}->SideLengths[Evaluate[t]]])],
PointLabels[Join@@{Transpose[{VertexPoints[t],{"A","B","C"}}],
{{l[[2]],"P"}}
},
LabelOffset->.03
]
}],AspectRatio->Automatic,PlotRange->All]
]