If M is the midpoint of CD, then the  is cos ABM = 8 / (235) sqrt

The Platonic solid known as the regular tetrahedron has four polyhedron vertices, six polyhedron edges, and four equal equilateral triangular faces. It is frequently referred to as “the” tetrahedron. Also included are the Wenninger model and uniform polyhedron.

1: According to a rule of trigonometry, the square of a side in a plane triangle is equal to the sum of the squares of the other sides minus twice that much of the product of those sides and the cosine of the angle between them.

To make things easier, set the tetrahedron’s side equal to 1.

A  = ( -1/2 , 0 , 0)

B  = ( 1/2, 0, 0  )

C =  (0, sqrt (3)/2, 0)

D =  (0, sqrt (3)/4 , 6/sqrt (3) )

M =  (0 , (3/8) sqrt (3) , 3/sqrt (3) )  =  (0, (3/8)sqrt (3) , sqrt (3) )

AB  = 1

AM = BM   =  sqrt   [ (-1/2)^2  + (3sqrt (3) / 8)^2  + (sqrt 3)^2   =  sqrt [ 1/4 + 27/64 + 3 ]   = sqrt (235) / 8

By the Law of Cosines

AM^2  =  AB^2 + BM^2 – 2 (AB) (BM)cos (ABM)

0 = 1 – 2 (1) (235 squared / 8) cos ABM

0 = 1 – [sqrt ( 235) / 8] since ABM

cos ABM = -1 / – [ sqrt (235) / 8]

8 / (235) sqrt = cos ABM

If M is the midpoint of CD, then the  is cos ABM = 8 / (235) sqrt

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