Stewart's Theorem

stewart.html

Exploration

Graphics saved as "pts10.eps".

Show that for any ΔABC as shown in the figure the relationship between the lengths of the labeled line segments is given by

.

Approach

Without loss of generality, place the triangle in a convenient position and use the distance formula repeatedly to verify the relationship.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Create points A, B, C and D in a convenient position.

Clear[c,m,x,y];
ptA=Point2D[{0,0}];
ptB=Point2D[{c,0}];
ptC=Point2D[{x,y}];
ptD=Point2D[{m,0}];

Compute the distances between the points.

a=Distance2D[ptB,ptC];
b=Distance2D[ptA,ptC];
d=Distance2D[ptC,ptD];

Verify that the relationship is an identity.

a^2*m+b^2*n-c*(d^2+m*n) /. n->c-m //Expand