Circle Inscribed in a Right Triangle

rttricir.html

Exploration

Show that if r is the radius of a circle inscribed in a right triangle with sides a and b and hypotenuse c, then r=(a+b-c)/2.

Approach

Position the triangle so that the sides of length a and b align with the x- and y-axes and the vertex opposite the hypotenuse is at the origin. Create the circle inscribed in this triangle and examine its radius.

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

The radius of the inscribed circle is found here.

Clear[a,b];
r1=FullSimplify[Radius2D[
    Circle2D[
     Triangle2D[{a,0},{0,b},{0,0}],
     Inscribed2D]],Assumptions->{a>0,b>0}]

Simplify the expression for the radius.

Clear[c];
Simplify[r2=r1 //. {a^2+b^2->c^2},Assumptions->{c>0}]

Since and r are equal.