Consider the following picture of the regular compound of five tetrahedra.
Does the line $CD$ exactly intersect the line of intersection of triangles $A$ and $B$?
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1 Answer
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Note that the vertices form a regular dodecahedron. We can use the following embedding of dodecahedron in $\mathbb R^3$ (image source: Wikipedia).
The coordinates of the vertices take a very nice form. In particular,
C = (0, -φ, 1/φ), D = (φ, 1/φ, 0),
Α1 = (1, -1, 1), Α2 = (1/φ, 0, -φ), Α3 = (0, φ, 1/φ),
Β1 = (φ, -1/φ, 0), Β2 = (-1/φ, 0, φ), Β3 = (-1, -1, -1),
where φ is the golden ratio. Let O = 1/2(φ, -1, 1/φ) be the midpoint of the line segment CD. It can be easily verified that
OA1 • (OA2 × OA3) = 0 and OB1 • (OB2 × OB3) = 0.
Therefore, O lies on both triangles A and B, so it must be on their line of intersection.
