WOLFRAM|DEMONSTRATIONS PROJECT

23. Construct a Triangle Given Two Sides and the Inradius

​
a
7.
b
5
r
1
show incircle
96.-12.
2
x
+
3
x
= 0
-32.-48.y+
3
y
= 0
This Demonstration draws a triangle
ABC
given two side lengths
a
and
b
and the inradius
r
(the radius of the inscribed circle). This construction involves solving a cubic and is not possible with a ruler and compass.
Let the third side length be
c
and let the area of
ABC
be
Δ
. Define the semiperimeter
s=(a+b+c)/2
. Since
(ra+rb+rc)/2
equals the sum of the areas of the three triangles with apex at the incenter,
Δ=rs
.
Using that together with Heron's formula,
Δ=
s(s-a)(s-b)(s-c)
gives
(s-a)(s-b)(s-c)=
2
r
s
, which is a cubic equation for
c
. The equations for
x=c
and
y=c-4
are shown at the bottom of the graphic.