I was wondering how to prove Euclid’s theorem: The medians of a triangle are concurrent. My work so far: First of all my interpretation of the theorem is that if a line segment is drawn from each of the 3 side’s medians to the vertex opposite to it, they intersect at one point. Since a […]

I have a rotated ellipse in parametric form: $$\begin{pmatrix}y \\ z\end{pmatrix} = \begin{pmatrix}a\cos t + b\sin t \\ c\cos t + d\sin t\end{pmatrix} \tag{1} $$ or, $$(y,z) = (a\cos t + b\sin t , c\cos t + d\sin t) \tag{2} $$ By using $$\cos^2 t + \sin^2 t = 1 $$ I can rewrite into: […]

What are interesting / non-trivial examples of smooth connected closed manifolds that are direct products or involve direct products? I am especially interested in orientable manifolds. Say, an $n$-torus $T^n$ is a direct product of $n$ copies of a circumference $S^1$. One can build a 3-manifold from a surface of genus $g$ as $M=M^2_g\times S^1$, […]

I am trying to understand the proof of the following claim: Let $f:A \subseteq \mathbb{S}^n \to \mathbb{S}^n$ be an $L$-Lipschitz* map (with $L <1$). Then $f(A)$ is contained in the interior of a hemisphere. *The distance on $\mathbb{S}^n$ can be either the intrinsic one or the extrinsic (Euclidean) one, it does not matter. In the […]

I want to learn explicitly proof of the proposition 9.2.3. Which books or lecture notes I can find? Please give me a suggestion. Thank you:)

I want to make the below picture into an egg with smooth surface. For the implementation in Mathematica, please, see this thread here. This thread considers mathematical methods to achieve the goal while the last one only in a single program. How can you make the 3D object smooth mathematically?

I have researched this question for days and can not locate a good answer. It could be a mathematical object that is defined by an axiom as Euclid or Hilbert. But if a curve is drawn between two points can it be should using only the rules of plane geometry that the curve is a […]

Suppose you’re given a circle with center $O$, I’m curious, how can one construct with ruler and compass three circles inside the larger circle such that each is tangent to the larger circle as well as to the other two?

An equidissection of a square is a dissection into non-overlapping triangles of equal area. Monsky’s theorem from 1970 states that if a square is equidissected into $n$ triangles, then $n$ is even. In 1968, John Thomas proved the following weaker statement: there is no equidissection of a unit square into an odd number of triangles […]

Given: $A = (0,0)$ $B = (0,-10)$ $AB = AC$ Using the angle between $AB$ and $AC$, how are the coordinates at C calculated?

Intereting Posts

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Do maximal proper subfields of the real numbers exist?
Proving a diagonal matrix exists for linear operators with complemented invariant subspaces
Stiefel-Whitney classes with Z-coefficients
If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex
Equation of sine wave around a circle
Looking for a source of an infinite trigonometric summation and other such examples.
Prove: $\int_0^1 \frac{\ln x }{x-1} d x=\sum_1^\infty \frac{1}{n^2}$
How to show that the nth power of a $nxn$ nilpotent matrix equals to zero $A^n=0$
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How would I undo a gradient function?
A square of a rational between two positive real numbers ?!
Inverse of $y=xe^x$
Geometric interpretations of $||z||_p = 1$?