Does expected triangle area change if a random point is added?

Question

Starting point

Case I

There are 3 random points in a volume. Calculate the expected area of the triangle.

Case II

Calculate the expected area of any of the 4 triangles that are formed if a 4th random point is added to the 3 random points from case I.

Goal

Prove or disprove that the expected area is the same for both case.

Assumptions

1. any volume in $\mathbb{R^3}$ can be selected that is suitable for the calculation (e.g. a cube or a ball or any other volume)
2. uniform random distribution in the volume
3. area is non-oriented

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The question is related to this question.

Answer 1

The expected area is the same because the joint distribution of the three points forming the triangle is the same in each case, namely the product of three independent uniform distributions over the volume.