Matematické Fórum

Skrytý text:

first thing to realize is that for given 4 points none of them will belong to the plane we look for;
if there was at least one such point, its distance from the plane is zero
and the need for equidistance will put all 4 points into this plane which makes them coplanar
so all the points lie outside the plane that we look for

another thing is that such plane would split the 3D space into two half-spaces
and each half-space contains at least one of the given 4 points, if they were all in one half-space,
that would necessarily make them coplanar because of equidistancy

so we can have two points in each half-space or one point in first one and 3 points in other one

let's see the case where points AB are in first half-space and CD in the other one
we can construct lines   and 
those two lines does not intersect as in such case the 4 points would be coplanar

the plane we look for has the normal    giving the equation 
where only is not known

now there exists for which the line lies in the plane
and also such that line q lies in the plane

planes are parallel since they share normal vector, let be

there is
, 
also

now, moving between constants gives us equations for planes parallel to
with
where the left side grows from to and the right side decreases from to
there is one plane there for which we get equality of the distances in the equation which gives us one and only one plane

as there are for ways how to distribute 4 points by two in each half-space, this gives us 4 plains


now let be the point in first half-space and points in the second half-space
those 3 points form the plane and for sure there is unique plane going through   parallel to
again there is one unique plane between those two where
as there are 4 ways to put only one of given point into the first half-space, this gives us 4 more planes

so the resulting number of planes should be 7

the assumption that no 3 of points are colinear is not necessary to add, collinearity of 3 points would make
the four points coplanar