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.