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Triangle Side Lengths from Altitudes

trisides.html

Exploration

Prove that the lengths of a triangle's sides whose altitudes are of lengths "trisides_1.gif", "trisides_2.gif" and "trisides_3.gif" are given by

"trisides_4.gif", "trisides_5.gif" and "trisides_6.gif"

where "trisides_7.gif", "trisides_8.gif" and "trisides_9.gif", and

"trisides_10.gif".

Approach

Construct a triangle with the formulas given for the side lengths and show that the altitude lengths are "trisides_11.gif", "trisides_12.gif" and "trisides_13.gif".

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

"trisides_14.gif"

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

"trisides_15.gif"

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

"trisides_16.gif"

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

"trisides_17.gif"

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}

"trisides_18.gif"


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