How long is the line between the center points of the opposite edges of a regular tetrahedron Let a tetrahedron ABCD, m be the midpoint of AB, n be the midpoint of CD, ab = 2, find Mn,

How long is the line between the center points of the opposite edges of a regular tetrahedron Let a tetrahedron ABCD, m be the midpoint of AB, n be the midpoint of CD, ab = 2, find Mn,

Firstly, it is proved that the opposite edges of regular tetrahedron are perpendicular to each other
Make ABC on the bottom of ah ⊥ and hang h to connect ch, and extend BD to f to connect AF,
∵AB=AC=AD,
H is the outer center of positive △ BCD,
⊥ CF ⊥ BD, (positive △ three in one)
F is the midpoint of BD,
∵△ abd is also a positive △,
∴AF⊥BD,
∵AF∩BD=F,
{BD ⊥ plane AFC,
∵ AC ∈ plane AFC,
BD ⊥ AC
Now let's turn to the main topic,
Take point E in AD and connect NE and me,
Then en and em are the median lines of △ DAC and △ abd, respectively,
∴EN//AC,ME//BD,EN=AC/2,ME=BD/2,
∵AC=BD,
∴EN=ME,
∵ BD ⊥ AC, (it has just been proved that the opposite edges are perpendicular to each other),
∴EN⊥ME,
The ∧ men is isosceles RT ∧,
∵AB=2,
∴ME=NE=1,
∴MN=√2ME=√2.