Relation between all 10 distance of 5 points

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SUMMARY

The discussion centers on the relationship between the distances of five points in Euclidean space, specifically focusing on points A, B, C, D, and a central point M. By specifying the six distances between four points (A, B, C, D), one can define a tetrahedron. Additionally, the distances from the central point M to the other points (AM, BM, CM) allow for the calculation of the distance DM, thus establishing a comprehensive understanding of the spatial relationships. The discussion references the Cayley-Menger determinant as a key mathematical tool for these calculations.

PREREQUISITES
  • Understanding of Euclidean geometry
  • Familiarity with tetrahedral geometry
  • Knowledge of distance formulas in three-dimensional space
  • Basic comprehension of determinants, specifically the Cayley-Menger determinant
NEXT STEPS
  • Study the properties of the Cayley-Menger determinant in depth
  • Explore advanced concepts in Euclidean geometry
  • Learn about distance calculations in multi-dimensional spaces
  • Investigate applications of tetrahedral geometry in computational geometry
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Mathematicians, geometry enthusiasts, and students studying spatial relationships in Euclidean space will benefit from this discussion.

Gerenuk
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If you specify all 6 distances between 4 points A, B, C, D you fix a tetraeder. If you now give three distances AM, BM, CM to a central point M you determine the position of that point within the tetraeder.
So theoretically one could calculate DM now?!

So my question is:
What is the relation between distances
AB, AC, AD, AM, BC, BD, BM, CD, CM, DM
in euclidean space?
 
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