WOLFRAM|DEMONSTRATIONS PROJECT

Right-Angled Tetrahedron

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a
1.2
b
1
c
1
show proof
Let
T=ABCD
be a tetrahedron with the three plane angles at
D
all right angles, that is,
∠ADB=∠BDC=∠CDA=90°
. (This is more explicitly known as a trirectangular tetrahedron.) Let
α=∠CAD
,
β=∠CBD
,
ϕ=∠ACB
. Then
cosϕ=sinαsinβ
. The lines that join the midpoints of opposite edges are equal and meet at a point. The proof, outlined in the Details, implies that these three lines are diagonals of a rectangular prism, intersecting at the center.