Relationships of Radii

We use the Radii Relationships in the Incircles and Excircles Wiki page. So: $\frac{1}{r} = \frac{1}{4} + \frac{1}{6} + \frac{1}{12} \\ \implies r = 2$ Now we use the area formula: $\mathrm{Area} = \sqrt{rr_1r_2r_3} = \sqrt{2 \times 4 \times 6 \times 12} = \boxed{24}$ For completeness, the $6, 8, 10$ triangle is such a triangle that exists.

Let us write the equations for the area of a triangle given one of its exradius:

$S = r_{A}(s - a)$

$S = r_{B}(s - b)$

$S = r_{C}(s - c)$

Firstly, let us derive a direct relationship between the area S and the semiperimeter s. Dividing the area by the radius, we obtain the following:

$\frac{S}{r_{A}} = (s - a)$

$\frac{S}{r_{B}} = (s - b)$

$\frac{S}{r_{C}} = (s - c)$

Adding the three equations above and noting that $a + b + c = 2s$ :

$\frac{S}{4} + \frac{S}{6} + \frac{S}{12} = 3s - 2s$

Thus: $\frac{S}{2} = s$ , or $S = 2s$

Now, taking the three original equations and multiplying them together yield:

$S^{3} = r_{A}*r_{B}*r_{C}*(s - a)(s - b)(s - c)$

Multiplying both sides by S:

$S^{4} = r_{A}*r_{B}*r_{C}*S*(s - a)(s - b)(s - c)$

Thus: $S^{4} = r_{A}*r_{B}*r_{C}*2s*(s - a)(s - b)(s - c)$

Since $S = \sqrt{s(s-a)(s-b)(s-c)}$ , we can write:

$S^{4} = r_{A}*r_{B}*r_{C}*2*S^{2} \rightarrow S^{2} = 4*6*12*2 = 576 \rightarrow S = 24$

Now that the conditions are set, we must prove that such triangle exists. From $S = 2s$ , we find that $s = 12$ ; now, using the first three equations, we have:

$24 = 4*(12 - a) \rightarrow a = 6$

$24 = 6*(12 - b) \rightarrow b = 8$

$24 = 12*(12 - c) \rightarrow c = 10$

This is a valid triangle, and a right one (note that $a^{2} + b^{2} = c^{2}$ ). Thus, the solution sought is that the area of the triangle is 24.

Realise that these are twice the ex-radii of the famous 3-4-5 right angled triangle, having ex-radii 2,3,6 respectively. We use scaling to understand that since linear dimension increased by a factor of 2 in the new triangle, area will increase by a factor of 4. So area is 4 times (1/2) (3) (4) which is 24.

same analysis