Constructing a triangle given 2 sides and the inscribed circle's radius A Triangle has 2 sides of length 4 and 9. The largest circle than can be drawn in the triangle so as to touch all 3 sides has radius 1 cm. Find the length of the 3rd side.
 A: There is one and only one circle that touches all 3 sides in a triangle, called the incircle. In fact, it is the largest circle that can be drawn within the triangle.
There is a simple area formula for a triangle, $A=rs$, where $s$ is the semiperimeter, and $r$ is the inradius [which in this case is $1$], which when combined with the Heron's formula gives:
$$r=1=\sqrt{\frac{(s-a)(s-b)(s-c)}{s}}$$
$$\text{So,  } s = (s-a)(s-b)(s-c)$$    
Now we have $s=\frac{4+9+c}{2}=6.5+\frac{c}{2}$. Let us dentone $\frac{c}{2}$ by $x$. So substituting: 
$$6.5+x = (6.5+x-4)(6.5+x-9)(6.5+x-c)$$
$$6.5+x = (x+2.5)(x-2.5)(6.5-x)$$
$$6.5+x = -x^3+6.5x^2+6.25x-40.625$$
Thus we have a cubic equation in $x=\frac{c}{2}$. In the next step, we multiply by $8$ to get it in terms of $c$:
$$x^3-6.5 x^2-5.25 x+47.125 = 0$$
$$c^3−13c^2−21c+377= 0$$
Which upon solving, we get three real roots, but only two positive, of which any can be a possible value of $c$, the remaining side:
$$c \approx 5.985522390888466479868150 \text{ or } c \approx 12.18399147026571219021056 $$
