# 23. Construct a Triangle Given Two Sides and the Inradius

23. Construct a Triangle Given Two Sides and the Inradius

This Demonstration draws a triangle given two side lengths and and the inradius (the radius of the inscribed circle). This construction involves solving a cubic and is not possible with a ruler and compass.

ABC

a

b

r

Let the third side length be and let the area of be . Define the semiperimeter . Since equals the sum of the areas of the three triangles with apex at the incenter, .

c

ABC

Δ

s=(a+b+c)/2

(ra+rb+rc)/2

Δ=rs

Using that together with Heron's formula, gives , which is a cubic equation for . The equations for and are shown at the bottom of the graphic.

Δ=

s(s-a)(s-b)(s-c)

(s-a)(s-b)(s-c)=s

2

r

c

x=c

y=c-4