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Question: The house that Jack built

Question:The house that Jack built

Markiyan Hirnyk 7714 Maple
May 10 2011
2

   We have four points A, B, C, E, which form a convex polygon in the plane Π, and a point S, which doesn't belong to  Π. The points A, B, C, E, and S form a pyramid. The following distances are measured with some error: |AB|=2.9, |BC|=4.43, |CE|=5.10, |EA|=5.2, |AC|=2.29, |BE|=3.17, |SA|=3.77, |SB|=4.27, |SC|=3.66, and |SE|=4.36. In fact, there doesn't exist a pyramid with exactly these distances. The problem arises: to find the numbers XAB, XBC, XCE, ... , and XSE such that the segments with these lengths form a pyramid and the sum (XAB-2.9)^2+(XBC-4.43)^2+...+(XSE-4.36)^2 is minimal. I think this can be solved as an optimization problem.
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