Distribution of an angle between 3 normally distributed 2D points

Let us treat $a=mu_A,b=mu_B,c=mu_C$ as complex numbers. Then we are interested in the random variable that is $Imlogfrac {(a+sigma_AX_A)-(b+sigma_BX_B)}{(c+sigma_C X_C)-(b+sigma_B X_B)}$ where $X_A,X_B,X_C$ are independent standard complex Gaussians (mean $0$, covariance $I_2$). The general case looks rather hopeless, but when $sigma$'s are small compared to the distances, we can linearize and get $$ Imleft{logfrac{a-b}{c-b}+frac{sigma_A}{a-b}X_A-frac{sigma_C}{c-b}X_C -sigma_Bleft[frac1{a-b}-frac1{c-b}right]X_Bright} $$ The first term is just the angle $alpha_0$ the means make and the sum of the last three terms is a complex Gaussian with $0$ mean and the covariance $sigma I_2$ where $$ sigma^2=sigma_A^2frac{1}{|a-b|^2}+sigma_C^2frac1{|c-b|^2}+sigma_B^2left|frac 1{a-b}-frac 1{c-b}right|^2 $$ (everything is measured in radians, of course). The projection to the imaginary axis yields the $N(alpha_0,sigma)$ normal law for the angle you are interested in.

This approximation breaks down for large variances, of course, so you may want to experiment a bit to see what the limitations are.