New answers tagged euclidean-geometry
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Are there infinitely many "generalized triangle vertices"?
There are uncountably many pseudovertices. Specifically, I will show that starting with the pseudovertex $X(4)$, we can make a small perturbation to a certain subset of the domain, propogate this ...
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Accepted
Constructing a polygon from another with collinearity constraints
Let $L_i$ be the bisector of $p_i$ and $p_{i+1}$, and let $f_i \colon L_i \to L_{i+1}$ be the central projection through $p_{i+1}$. This is a projective transformation with constructible coefficients. ...
- 5,967
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Finding angle with geometric approach
I do not know if this is elementary enough, though the exercise could fit in an Olympiad style easily. So here is another analytic way. In the figure, let the point $D$ be the center $(0;0)$ and ...
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-4
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Is Euclid dead?
I'm suggesting teaching the foundations of mathematics. I would pick a system like:
ETCS (Elementary Theory of the Category of Sets),
HoTT (Homotopy Type Theory),
Dependent Type Theory,
ZFC?
I ...
5
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Minimum diameter of set inscribed in a unit sphere
It is Jung's theorem. If the diameter would be less than
$d=\sqrt{\frac{2(n+1)}{n}}$, there would be a smaller enclosing ball.
https://en.wikipedia.org/wiki/Jung%27s_theorem
Supplement: In the plane, ...
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Minimum diameter of set inscribed in a unit sphere
Your conjecture that a regular simplex minimizes the diameter is correct.
Let $M$ be the intersection of $A$ with the boundary of the minimum ball containing $A$. Then the convex hull of $M$ contains ...
- 5,967
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Minimum diameter of set inscribed in a unit sphere
$\sqrt{2}$ is indeed a strict lower bound. A fairly short proof goes as follows:
Working with closed sets, we can assume $A$ to be a
compact set of the closed unit ball $B$ centered at $0$,
perhaps ...
- 15.6k
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