Hearted Youtube comments on Mathologer (@Mathologer) channel.
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Ah, I did it. Here is the proof of the cube shadow theorem (with terms defined at 13:50):
Without loss of generality, assume S is the lowest point of the cube. The area of the shadow is then just the area of the parallelograms formed by the projections of the faces spanned by (u, v), (v, w), and (w, u) onto the x-y plane, respectively. These areas are given by the z components of the cross products u x v, v x w, and w x u. By construction, these cross products are just the vectors w, v, and u, so the area is given by the sum of the z components of all three vectors.
For the length of the line projection, assume (without loss of generality) that S is at height z=0. Since we also assumed it is the lowest point, the distance between S and the opposite vertex u + w + v (which is the highest vertex by symmetry) is the length of the z projection. Therefore, this length is also the sum of the z components of u, v, and w.
This completes the proof.
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