Is it possible to prove that for all kinds of simple polygon, regardless of whether it is convex or concave and with no opening, the centroid of the polygon must ( or may not) lie inside the polygon? The wiki link above gives example of polygon which has the centroid lying outside the polygon: A […]

$O(2,3)$, $A(2,0)$, $B\left(1,\dfrac{1}{\sqrt{3}}\right)$ are the vertices of $\Delta{OAB}$ on the $\text{x-y}$ plane. Let $\text{R}$ be the region consisting of all points $P$ inside the triangle, which satisfy: $$d(P,OA)\geq \min\{d(P,OB), d(P,AB)\} $$ For a random distribution of point P, the probability that it lies in the region $\text{R}$ is of the form: $a-b\sqrt[c]{d}$ $\text{Find:}d^{a}+ c^{b}$$$$$ $d(X,YZ)$ […]

Absolute geometry (as I know the term) is just (Euclidean geometry) $-$ (parallel postulate). (It is sometimes also called neutral geometry, because it is “neutral” w.r.t. the parallel postulate.) The only spaces I know of which satisfy its axioms are Euclidean spaces and hyperbolic spaces, both of which are obviously Riemannian manifolds. Question: Are these […]

Question: Suppose points A and B are all on the circle C with center O. Prove that the perpendicular bisector of segment AB contains O. Here is how my proof goes. I used proof by contradiction. SO I assumed to the contrary that the perpendicular bisector does not pass through point O. WOLOG assume it […]

What is the geometric meaning of Cauchy’s functional equation? $$f(x+y) = f(x)+f(y) \quad \forall x,y$$

I have to cover an equilateral triangle (whose sides are 1m long) with 5 identical circles: what’s the minimum radius of the circles?

What is this geometric pattern called?

Problem Prove that if $\mathbf{y}:E\to M$ is a proper patch, then $\mathbf{y}$ carries open sets in $E$ to open sets in $M$. Deduce that if $\mathbf{x}:D \to M$ is an arbitrary patch, then the image $\mathbf{x}(D)$ is an open set in $M$. (Hint: To prove the latter assertion, use Cor 3.3.) Finally, prove that every […]

What is the polar form for a superellipse with semidiameters $a$ and $b$, centered at a point $(r_0, θ_0)$, with the $a$ semidiameter at an angle $\varphi$ relative to the polar axis?

I suppose this geometric shape is something very ‘special’. I cannot clarify in short about being ‘special’, but I think this shape stands together with such special shapes like the square and the regular hexagon. Here it is: So it is a parallelogram based upon a square. Its height is equal to the side of […]

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