Let ra, rb, and rc be the radii of the excircles opposite vertices A, B, and C respectively with 0 ≤ ra ≤ rb ≤ rc.
From equation (7) at the link,
r(rarb + rbrc + rcra) = sΔ = s(rs)
or
s2 = rarb + rbrc + rcra = 2*3 + 3*4 + 4*2 = 26
, where r, s, and Δ are the inradius, semiperimeter, and area of ΔABC respectively.
The largest angle in ΔABC is opposite the largest excircle. Let T be the point of tangency of this excircle with sideline CA.
It is easy to show that |CT| = s.
Since the center of this excircle lies on the bisector of angle C,
Angle C = 2*arctan(rc/s) = 2*arctan(4/√26) ~= 76.225853 degrees.
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