Ex-circles In-circle

Let $\Delta$ be the area of the triangle, $s$ be the semi-perimeter, $r$ be the inradius, and let $r_a, r_b, r_c$ be the radii of the excircles. Then the following relations hold; see here, for reference.

$r = \dfrac{ \Delta } {s }$

$s^2 = r_a r_b + r_a r_c + r_b r_c$

and

$\Delta = \sqrt{ r r_a r_b r_c }$

Substitute the first relation into the third, to get,

$r s = \sqrt{ r r_a r_b r_c }$

Squaring, gives,

$r s^2 = r_a r_b r_c$

Using the second relation, we obtain,

$r = \dfrac{ r_a r_b r_c }{ r_a r_b + r_a r_c + r_b r_c }$

Substituting the given values for $r_a , r_b , r_c$ gives,

$r = 2.235868051177 \approx 2.236$