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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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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. ...
1 vote

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 votes

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 votes

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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0 votes

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 ...
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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 ...

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