Is sum of expected triangle areas equal to expected area of triangle sums?

Question

Starting point

Given are 4 multinormal distributions $\mathcal{N}(\vec{\mu}_1,\Sigma), \mathcal{N}(\vec{\mu}_2,\Sigma),\mathcal{N}(\vec{\mu}_3,\Sigma),\mathcal{N}(\vec{\mu}_4,\Sigma)$ in $\mathbb{R^3}$. From every distribution a random point is selected. The expected areas of the 4 triangles that can be formed with the 4 random points are $\mathbb{E}(A_1),\mathbb{E}(A_2),\mathbb{E}(A_3),\mathbb{E}(A_4)$. The expected area of the sum of the 4 triangle areas is $\mathbb{E}(A_1+A_2+A_3+A_4)$.

Goal

Prove or disprove that $\mathbb{E}(A_1+A_2+A_3+A_4)=\mathbb{E}(A_1)+\mathbb{E}(A_2)+\mathbb{E}(A_3)+\mathbb{E}(A_4)$

Assumptions

1. the covariance matrix is $\Sigma=\sigma^2\mathbb{\bf{I}}$ (with $\mathbb{\bf{I}}$ the identity matrix in $\mathbb{R^3}$ and $\sigma^2$ the variance)
2. triangle points and distributions have a one-to-one correspondence (i.e. no 2 points are from the same distribution)
3. area is non-oriented

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Answer 1

This is true quite generally and is called the linearity of expectation. It has nothing to do with the particulars of your problem.