Little explorations with HP calculators (no Prime)

04072017, 05:19 PM
(This post was last modified: 04072017 05:50 PM by Han.)
Post: #136




RE: Little explorations with the HP calculators
(04072017 12:57 PM)SlideRule Wrote:(04072017 02:29 AM)Gerson W. Barbosa Wrote: ...The shape of the outer triangle is irrelevant ... I don't think we can assume that x=y=z (which is the implication in assuming the sides are 2x, 3x, 4x). EDIT (Additional comments): The \( n \)dimensional volume of an object in \( \mathbb{R}^n \) (Euclidean space) is invariant under unimodular transformations (i.e. transformations of the form \(\mathbf{x} \to \mathbf{A} \cdot \mathbf{x} \) with \( \mathbf{A} =1 \)). So volume (in this case, 2dimensional "volume" is really area) remains the same; but lengths are not preserved. On the other hand, relative positions are preserved. For example, since F is 1/4 the distance of BC from the point B, it will remain 1/4 the distance from B under any unimodular transformation, and B, F and C will remain collinear. The assumption that the outermost sides are 2x, 3x, and 4x, while in my opinion incorrect, do not affect the final result because (in the end) we are looking at ratios of lengths (i.e. relative positions of points on a line segment). 

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