## New answers tagged euclidean-geometry

3
votes

### An inequality in an Euclidean space

It doesn't hold, although it looks quite rare. For example $$u:=\begin{pmatrix}1\\1\\1\end{pmatrix}, \quad v:=\begin{pmatrix}\;1\\\;1\\27\end{pmatrix}, \quad x:=\begin{pmatrix}-1\\\;\;6\\\;25\end{...

7
votes

Accepted

### An inequality in an Euclidean space

Imagine that $v$ is (almost) equal to $(1,0,\ldots,0)$, then for $x=(x_1,\ldots,x_n)$, $\sum x_i=:na$, we should check that $(x_1-a,x_2-a,\ldots,x_n-a)\cdot(0,x_2,\ldots,x_n)\geqslant 0$, in other ...

1
vote

### An inequality in an Euclidean space

There is nothing special about $\mathbf{u}=(1,\dots,1)$. I work with an arbitrary vector in $(0,\infty)^n$.
Claim) Let $n\geq 3$ and $\mathbf{u}=(u_1,\dots,u_n),\mathbf{v}=(v_1,\dots,v_n)\in
(0,\...

2
votes

### What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

The basic fact is that one can express the required extreme values of the areas in the generic case (our use of this term is explained below). The simple formulae involved can be found below. To do ...

5
votes

### What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

I argue that the expression for infimum is more complicated and cannot be given by a single formula unless certain inequalities involving $a,b,c,d$ are assumed.
Let us work in the setting that OP ...

15
votes

### What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

Another proof that the area is maximized when the quadrilateral is cyclic is the following:
First we consider a cyclic quadrilateral with the given sides lengths (the existence of such a quadrilateral ...

19
votes

### What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

The area, call it $K$, is indeed maximized when the quadrilateral is cyclic,
in which case $K$ is given by Brahmagupta's
remarkable generalization of Heron's formula:
$K = \sqrt{(s-a)(s-b)(s-c)(s-d)}$ ...

2
votes

### Why do all incidence theorems follow from Pappus' theorem?

There's a recent talk by Sergey Fomin about this topic exactly - https://www.youtube.com/watch?v=kU5PazTfImo
Abstract: We show that various classical theorems of real/complex linear incidence geometry,...

2
votes

### Inscribing one regular polygon in another

Here is a proof for large enough coprime $m$ and $n$.
I use some basic properties of cyclotomic polynomials, be free to ask for details if needed. In particular, I use that the sum of roots of $\Phi_k$...

Top 50 recent answers are included

#### Related Tags

euclidean-geometry × 503mg.metric-geometry × 268

plane-geometry × 95

discrete-geometry × 50

reference-request × 45

triangles × 45

polygons × 27

elementary-proofs × 26

convex-geometry × 22

ho.history-overview × 21

convex-polytopes × 19

projective-geometry × 19

ag.algebraic-geometry × 16

linear-algebra × 16

computational-geometry × 16

dg.differential-geometry × 15

pr.probability × 15

gr.group-theory × 15

nt.number-theory × 14

co.combinatorics × 13

real-analysis × 13

inequalities × 13

polyhedra × 13

algorithms × 11

terminology × 11