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Plane Geometry is about flat shapes like lines, circles and triangles , shapes that can be drawn on a piece of paper

3 votes
1 answer
178 views

Analytic or holomorphic extension of the ellipse perimeter function

Let ${\mathbb{R}^2}^+=\{(x,y)\in \mathbb{R}^2\mid x>0, y>0\}$. Let $P:{\mathbb{R}^2}^+\to \mathbb{R}$ be the function with $P(a,b)=$ $\text{The perimeter of ellipse}\;\; \frac{x^2}{a^2}+\frac{y^2}{b^ …
Ali Taghavi's user avatar
5 votes
2 answers
560 views

Geometry of Level sets of elliptic polynomials in two real variables

Updated: A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its last homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.The two answers to this post provide a …
Ali Taghavi's user avatar
1 vote
1 answer
125 views

An asymptotic version of the Isoperimetric inequality

Let $U$ be a simply connected bounded open set in $\mathbb{R}^2$. The area of $U$ is denoted by $A$. (We do not assume any thing about its boundary). Assume that $\gamma_n$,s are smooth simple clos …
Ali Taghavi's user avatar
3 votes
1 answer
137 views

A geometric property about certain polynomials in two variables

Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$ where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last …
Ali Taghavi's user avatar
9 votes
1 answer
376 views

Are two triangles with equal corresponding medians, congruent?

Is the hyperbolic or spherical analogy of the following Euclidean fact, true? Two triangles with equal corresponding medians are congruent. More precisely: Assume that $\Delta ABC$ and …
Ali Taghavi's user avatar
3 votes
2 answers
284 views

Can the "Bisector" be represented by a holomorphic function?

Note: In this question, a complex number is counted as a vector initiated from the origin. ______________________________________________________________- Is there a holomorphic function $B:\mat …
Ali Taghavi's user avatar
1 vote

Meeting a set of lines in $\mathbb{R}^n$

For $n=2$ we define $M$ as follows: $M$ is the union of the following sets: 1)The intersection with $x\_$ axis for lines not parallel to this axis. 2)The intersection with $y\_$axis for lines perp …
Ali Taghavi's user avatar
0 votes
1 answer
83 views

Hyperbolic version of Sylvester co-linear problem

Is the hyperbolic version of Sylvester co linear problem true?
Ali Taghavi's user avatar
2 votes
1 answer
613 views

Half spaces free of roots of a given polynomial

I thank Loic Teyssier and Emil Jerabek who helped me to revise the two previous version This question is motivated by the following fact in complex variable:(I learned this fact from the book of …
Ali Taghavi's user avatar
4 votes
0 answers
352 views

A generalized ellipse

We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$ where $a,b$ are two given points in the plane and $\lambda$ is a constant. Now we consider the follo …
Ali Taghavi's user avatar
3 votes
1 answer
161 views

Polygons with centroid at origin and vertices on compact codimension one submanifold of $\ma...

Let $M$ be a compact codimension one submanifold of $\mathbb{R}^{n}$ which does not contaion $0$ and the origin lies in the bounded component of$\mathbb{R}^{n}-\{0\}$. Is it true to say that: F …
Ali Taghavi's user avatar
2 votes
1 answer
357 views

Perimeter of ellipse: Combination of two geometries

Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ such that for every ellipse $\gamma$ in the plane we have:$$\text{The Euclidien perimeter of}\; \gamma=\lambda (g\text{-diameter of}\;\gamma)$ …
Ali Taghavi's user avatar
2 votes
2 answers
181 views

A quantity associated to a triangle

Let $\Delta ABC$ be a triangle in the plane. Let $P_{1}, P_{2}, P_{3}$ be the intersection points of bisectors, medians and altitudes, respectively. We define the quantity: \begin{equation} Q(\Delta …
Ali Taghavi's user avatar
2 votes
1 answer
246 views

The points of half area of a triangle

Let $S$ be a simply connected Riemannan surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is calle …
Ali Taghavi's user avatar