Let
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
be a regular tetrahedron. To an arbitrary point
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
on one edge, say
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
, corresponds the point
![P = P(M)](/media/m/9/a/e/9ae20be5d44ee91e5a001c1f31e13d61.png)
which is the intersection of two lines
![AH](/media/m/1/7/0/1700da59d12a188862b9dc234aba8941.png)
and
![BK](/media/m/b/9/c/b9cf4d2548fb2edc8dc197004ed51cf4.png)
, drawn from
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
orthogonally to
![BM](/media/m/9/b/a/9ba306de3378f6c32d1ba470951ff4a4.png)
and from
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
orthogonally to
![AM](/media/m/9/2/1/921d54bb92ada2d2120b2591b722ea12.png)
. What is the locus of
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
when
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
varies ?
%V0
Let $ABCD$ be a regular tetrahedron. To an arbitrary point $M$ on one edge, say $CD$, corresponds the point $P = P(M)$ which is the intersection of two lines $AH$ and $BK$, drawn from $A$ orthogonally to $BM$ and from $B$ orthogonally to $AM$. What is the locus of $P$ when $M$ varies ?