Author Topic: Some geometric problem in three dimensions.  (Read 1703 times)

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Offline Horrowind

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Some geometric problem in three dimensions.
« on: September 17, 2011, 04:57:16 pm »
okay, so here is a problem I have long been thinking of. Its basically an extension to an (probably) well known problem, taking it from one dimension to another.

First the two-dimensional version, a warming up so to speek:
Given three concurrent lines on a plane (so they have all have a point in common), find an equilateral triangle so that on each line is a corner of the triangle.
The solution is rather simple, if the trick for solving it is known. Overwise its quite hard :) (Tip: rotate)

Okay, so here the more complex versions of the problem:

Given three concurrent lines in space find such an triangle.
(sadly rotating is not as easy when there is no trivial axis to rotate)
and

Given four points in space, what are the restrictions for a point and a regular tetrahedron, so that the projection of the vertices of tetrahedron from this point results in these four points. Or the other way around: Is it possible to construct a regular tetrahedron so that the four lines, connecting one point of these four given points with one vertex of the tetrahedron, intersect in one point, and what are the restrictions of this problem.

Note that the first of these problems is equivalent to solve this equation system:
a2 + b2 + a b k1 = 1
b2 + c2 + b c k2 = 1
c2 + a2 + c a k3 = 1
for given k1, k2, k3 (simply set the size of the triangle to one and use the law of cosines).

Inputing this into maple gives some horrifying results  *.* (okay, im a n00b with maple, but still).

Maybe someone has an idea how to solve this.