Articles of geometry

Prove ( or disprove) that for all kinds of simple polygon, the centroid lies inside the polygon

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 […]

Geometrical Probability

$O(2,3)$, $A(2,0)$, $B\left(1,\dfrac{1}{\sqrt{3}}\right)$ 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}$$$$$ $d(X,YZ)$ […]

Is there a way to prove that absolute geometry must take place on a Riemannian manifold?

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 […]

inscribed triangle with circle

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 […]

Geometric meaning of Cauchy functional equation

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

Geometry – Equilateral triangle covered with five circles

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?

What is this geometric pattern called?

Proving that every patch in a surface $M$ in $R^3$ is proper.

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 […]

Polar form of a superellipse?

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?

Very special geometric shape – parallelogram (No name yet?)

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 […]