Date: Mar 5, 2006 5:53 PM
Author: Rich Kleinschmidt
Subject: Re: A few tetrahedron (in 3 space) questions
A purely algebraic approach would be to assume that the center is equidistant from all three vertices. So if (p,q,r) is the center, use the distance formula to find the distance to each vertex. For instance the distance to (1, 0, 1) would be sqrt( (p-1)^2 + (q-0)^2 + (r-1)^2 )
Once you have the 4 distances, pick one and write three equations setting it equal to each of the other three vertices. Now you have a system of three equations with three variables. Solve it.
-------------- Original message ----------------------
From: Patrick <email@example.com>
> > I'm not sure if i'm posting in the proper place - i
> > needed some help with a question
> > The four vertices of the tetrahedron (0,0,0) (1,1,0)
> > (1,0,1) and (0,1,1)
> > 1. verify that the four points listed are all
> > equidistant from each other. I'm sure if you find the
> > lengths of all the sides this could show that all
> > points are equal distances from each other.
> > 2. Find the centre of the given regular tetrahedron?
> > This is the question i'm having trouble with - i know
> > that the three coordinates are all equal and the
> > centre is the same distance from each of the
> > verticies. (but i'm not sure where to go from there?
> take the first three points you gave. they form a triangle - one of the faces of
> the tetrahedron. find that triangle's centre. let's say it's at (a, b, c). now
> consider the line passing through (a, b, c) and the fourth point/vertex (0, 1,
> 1). the centre is somewhere on that line. to find it, you'll have to repeat the
> process with another face. let's say you use the last three points and find the
> centre of that triangle to be (d, e, f). then you consider the line passing
> through (d, e, f) and the fourth point/vertex (0, 0, 0). the centre will be on
> this line as well. from these two lines, how could you find the centre?
> > 3.use the dot product methods to find the angle
> > formed between any two vectors extending from the
> > centre of the regular tetrahedron to two of its
> > vertices.
> > huh?
> let's say you found the centre to be (p, q, r). then you can find a vector from
> (p, q, r) to a vertex, say (0, 0, 0). find a second one and calculate their dot
> product. that will help you solve for the angle using the dot product equation.
> will the angle be the same for any two vertices you choose?
> > I'd appreciate any help - or advice on how i should
> > approach this problem. THANKS !
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