Triangle Radii

trirad.html

Exploration

Prove that the radius, r, of a circle inscribed in a triangle is given by

where , and , and are the lengths of the triangle's sides.  Furthermore, prove that the radius, R, of the circle circumscribing the triangle is given by

.

Approach

Construct a triangle with the given side lengths.  Construct the associated inscribed and circumscribed circles and examine the radius of these circles.

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[s1,s2,s3,S,P,e1,e2];
T=(Triangle2D[{s1,s2,s3}] //FullSimplify) //.
     {Sqrt[-e1_/s_Symbol^2]:>Sqrt[-e1]/s}

Construct the inscribed circle and compute its radius.  Simplify the resulting expression using appropriate substitutions.

FullSimplify[Radius2D[Circle2D[T,Inscribed2D]],Assumptions->{s1>0,s2>0,s3>0}]

Construct the circumscribed circle and compute its radius.  Simplify the resulting expression using appropriate substitutions.

FullSimplify[
     Radius2D[Circle2D[T,Circumscribed2D]],
     Assumptions->{(-s1+s2+s3)*(s1-s2+s3)*(s1+s2-s3)==P,s1>0,s2>0,s3>0}] /.
     s1+s2+s3->S