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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
2
votes
2
answers
181
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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 …
0
votes
1
answer
83
views
Hyperbolic version of Sylvester co-linear problem
Is the hyperbolic version of Sylvester co linear problem true?
1
vote
1
answer
125
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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 …
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 …
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 …
2
votes
1
answer
613
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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 …
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 …
2
votes
1
answer
246
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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 …
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:$$\text{The Euclidien perimeter of}\; \gamma=\lambda (g\text{-diameter of}\;\gamma)$ …
3
votes
1
answer
161
views
Polygons with centroid at origin and vertices on compact codimension one submanifold of $\ma...
Let $M$ be a compact codimension one submanifold of $\mathbb{R}^{n}$ which does not contaion $0$ and the origin lies in the bounded component of$\mathbb{R}^{n}-\{0\}$.
Is it true to say that:
F …
3
votes
1
answer
137
views
A geometric property about certain polynomials in two variables
Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$
where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last …
4
votes
0
answers
352
views
A generalized ellipse
We know that an ellipse is the locus of all point $z$ in the plane with $$|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 …
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^ …
5
votes
2
answers
560
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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 …