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        riddles >> medium >> Points and Planes
        
(Message started by: THUDandBLUNDER on Sep 24^th, 2003, 6:35am)

Title: Points and Planes
Post by THUDandBLUNDER on Sep 24^th, 2003, 6:35am

Given four points in R³, not all in the same plane, how many different planes are there that are equidistant from all four points?

Title: Re: Points and Planes
Post by towr on Sep 24^th, 2003, 8:36am

I get [hide]7[/hide]
But don't ask me to prove it ([hide]I just used a tertrahedron I have lying around[/hide])

Title: Re: Points and Planes
Post by James Fingas on Sep 24^th, 2003, 9:40am

It seems to me:
[hide]
There are two classes of solution: pick three points to form a plane. Measure the distance of the fourth point to the plane, calling it d. Then put spheres of radius d/2 around all four points. There is obviously a plane which touches all four spheres.

There are four solutions of this class (with three points on one side, and one point on the other).

The second class of solution involves picking two points, joining them with a line. Join the other two points with a line as well. Measure the distance between the lines, calling it d. Put two cylinders of radius d/2 with axes on the lines. There is obviously a plane tangent to both cylinders.

There are three solutions of this class (with two points on each side).

Since no plane can have more zero points on one side (or the four points would have to be coplanar), and for each choice of which points are together on one side, there is a unique plane (probably provable using my constructions above), and none of these planes can be the same as any of the other planes (provable using the constructions above), then there are exactly 7 planes, just like towr says.[/hide]