Consider a triangle \( \mathrm{ABC} \) and let \( \mathrm{a}, \math...

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Consider a triangle \( \mathrm{ABC} \) and let \( \mathrm{a}, \mathrm{b} \) and \( \mathrm{c} \) denote the lengths of the sides opposite to vertices
P \( A, B \) and \( C \) respectively. If \( a, b, c \) (in order) are in arithmetic progression such that \( x^{2}+3 x+5=0 \)

W and \( 3 x^{2}+a x+c=0 \) have a common root, then find the radius of the incircle of triangle \( A B C \).
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