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A line is also considered a degenerate rectangle (or square). I guess any rectangle with a zero side. Ohh, gotcha, thank you so much. Seemed pretty insignificant, but definitely important. Thanks again. So in this context it means specifically that A, B, C A, B, C and D D are four distinct points.
Jan 11, 2023 · Let (g, [⋅, ⋅], δ) be a triangular Lie bialgebra with classical r -matrix r ∈ ⋀2g. In such a case we say that r is a triangular structure on g. A triangular structure r on g is said to be non-degenerate if it induces non-degenerate map r _: g ∗ g. This definition is given by Pavel Etingof in his lecture notes on compact quantum groups.
Oct 10, 2023 · $\begingroup$ By "non-degenerate triangle" the author may merely mean a triangle that three defined angles, none of which is zero. In standard analysis, such a triangle must have non-zero area, but on this page we are accepting many things that are impossible in standard analysis, such as two points that are indistinguishable but do not coincide.
To check whether 3 3 vectors form a non-degenerate triangle, i.e. a triangle with positive area, you only need to check if the sum of them is 0 0 and the are not colinear. 3 3 vectors that sums 0 0 can be written as AB→ +BC→ +CA→ = 0 A B → + B C → + C A → = 0 so AB→ +BC→ = −CA→ A B → + B C → = − C A →. The are not ...
Mar 19, 2023 · 1. As far as I am concerned the concept of a degenerate triangle wasn‘t invented, but rather is a consequence of being not careful enough when trying to find a mathematically rigorous definition of a triangle. For example the convex hull of three points also encapsulates degenerate cases like some points coinciding or all points being collinear.
Apr 13, 2017 · Degenerate means some kind of element of a class belongs to a simpler class. In geometry, if a rectangle or triangle has side length 0, then it's degenerate. Similarly, if a circle has radius 0 it's degenerate. In graph theory, a graph is degenerate if every vertex has at most d neighbors later than it in ordering, something like that here.
Aug 11, 2024 · Reciprocal triangle inequality: In any non-degenerate triangle, the sum of the reciprocals of any two sides is less than the sum of the reciprocal of the third side and half the reciprocal of the inradius.
My thoughts are first there will be 25 ∗ 16 ∗ 9 25 ∗ 16 ∗ 9 cases in total, if we dont count when the triangle is degenerated. It is the permutation so we divide by 3! 3! to get 600. A straight line will make the triangle degenerate, so we need to minus those cases. There are 5 points on one diagonal of the square, 4 on its left and ...
In some mathematical English, a triangle means exactly what it did 2,300 years ago: "A triangle is plane figure bounded by three straight lines." (A Text-book of Euclid's Elements, Hall & Stevens, 1888). This definition seems unclear. But later, Proposition 20 clarifies it (emphasis added):
May 21, 2018 · $\begingroup$ If we are told that three sides of a (non-degenerate) triangle (without any other information about the triangle) are $3,4,x$ then we learn from the triangle inequality that $1<x<7$. We are however told an additional piece of information: that the triangle is acute. $\endgroup$ –
