= Solution
Under independent <normal distributions>, <maximum likelihood estimation> minimizes $(y_A-\alpha)^2+(y_B-\beta)^2+(y_C-\gamma)^2$ subject to $\alpha+\beta+\gamma=\pi$. A <Lagrange multiplier> gives the same correction at every corner. Writing $s=y_A+y_B+y_C$, the interior estimates are
$$
\boxed{\widehat\alpha=y_A+\frac{\pi-s}{3},\quad\widehat\beta=y_B+\frac{\pi-s}{3},\quad\widehat\gamma=y_C+\frac{\pi-s}{3}.}
$$
The result is unchanged if the common <variance> is unknown, since profiling it still minimizes the residual sum of squares. If physical positivity of the three angles is imposed and this projection gives a negative component, the constrained estimate over the closed simplex is instead $\widehat\theta_i=\max(y_i-\lambda,0)$, with $\lambda$ chosen to make the components sum to $\pi$. If only strictly nondegenerate triangles are admitted, a boundary optimum is merely an unattained likelihood supremum.
The normal-error model is a useful small-error approximation, but its unbounded tails assign positive probability to impossible measured angles. Periodicity or a measurement convention may matter for large errors, and equal <variance> and independence require experimental justification. The exact sum constraint on the true angles does not itself invalidate independence of three raw measurement errors. After imposing that constraint, however, the fitted-angle errors are correlated: the covariance of the interior estimator is $\sigma^2(I-\boldsymbol1\boldsymbol1^T/3)$.
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