Triangle Side Lengths from Altitudes

trisides.html

Exploration

Prove that the lengths of a triangle's sides whose altitudes are of lengths , and are given by

, and

where , and , and

.

Approach

Construct a triangle with the formulas given for the side lengths and show that the altitude lengths are , and .

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

Construct the triangle with the given side lengths.

Clear[h1,h2,h3,H1,H2,H3,H];
T=Triangle2D[2*{h1*H1^2,h2*H2^2,h3*H3^2}/H] //FullSimplify

Compute the lengths of each altitude (squared), which is the distance from the vertex to the opposite side.

alts1={
   Distance2D[Point2D[T,1],Line2D[T,2,3]]^2,
   Distance2D[Point2D[T,2],Line2D[T,1,3]]^2,
   Distance2D[Point2D[T,3],Line2D[T,1,2]]^2} //FullSimplify

A few substitutions verify that the altitude lengths (squared) are the expected values.

alts2=alts1 //. {H1->h2*h3,H2->h1*h3,H3->h1*h2} //FullSimplify //Factor

alts3=alts2 //. {h2*h3->H1,h1*h3->H2,h1*h2->H3}

alts4=alts3 /. {
   ( H1-H2-H3)(H1+H2-H3)( H1-H2+H3)(H1+H2+H3)->-H^2,
   (-H1-H2+H3)(H1-H2+H3)(-H1+H2+H3)(H1+H2+H3)->-H^2}