 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 areThe 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 ${\hat{\theta }}_i=\max (y_i-\lambda ,0)$, 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 ${\sigma }^2(I-11^T/3)$.