Recent questions in Elementary geometry

Midpoint Formula
Answered

Waldronjw
2022-06-30

$(1,1),(4,5)$

Euclidean Geometry
Answered

Banguizb
2022-06-30

Angle Bisectors
Answered

Reginald Delacruz
2022-06-30

Properties of parallelograms
Answered

gnatopoditw
2022-06-30

Postulates
Answered

DIAMMIBENVERMk1
2022-06-29

Axiom 1.2.1 (Peano Postulates). There exists a set $\mathbb{N}$ with an element $1\in \mathbb{N}$ and a function $s:\mathbb{N}\to \mathbb{N}$ that satisfies the following three properties.

a. There is no $n\in \mathbb{N}$ such that $s(n)=1$.

b. The function s is injective.

c. Let $G\subseteq \mathbb{N}$ be a set. Suppose that $1\in G$, and that $g\in G\Rightarrow s(g)\in G$. Then $G=\mathbb{N}$.

Definition 1.2.2. The set of natural numbers, denoted $\mathbb{N}$, is the set the existence of which is given in the Peano Postulates.

My question is: From my understanding of the postulates, we could construct an infinite set which satisfies the three properties. For example, the odd numbers $\{1,3,5,7,\dots \}$, or the powers of 5 $\{1,5,25,625\dots \}$, could be constructed (with a different $s(n)$, of course, since $s(n)$ is not defined in the postulates anyway). How do these properties uniquely identify the set of the natural numbers?

Midpoint Formula
Answered

Lucille Cummings
2022-06-29

Postulates
Answered

dourtuntellorvl
2022-06-29

$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots $

and the sum of the reciprocals of the primes

$\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots $

are divergent, while the sum

$\sum _{n=0}^{\mathrm{\infty}}\frac{1}{{n}^{p}}$

is convergent for all $p>1$. This would lead one to conjecture something like:

For all $\u03f5>0$, there exists an $N$ such that if $n>N$, then there exists a prime between $n$ and $(1+\u03f5)n$.

Question: Is this conjecture true? If it is true, is there an expression for $N$ as a function of $\u03f5$?

Midpoint Formula
Answered

landdenaw
2022-06-29

Properties of parallelograms
Answered

Quintin Stafford
2022-06-27

In the method, I took $\mathrm{\u25b3}ABC$ with $D$, $E$, and $F$ as midpoints of $AB$, $AC$, and $BC$, respectively; and I joined $DE$ and $EF$ so that I get a parallelogram $\u25fbDEFB$.

I know what the answer is because one can easily make that out. Also, those four triangles (four because the parallelogram can still be divided into two triangles and the rest two triangles add up to four) so it's simple that the area of $\u25fbDEBF$ will be 1/4 of $\mathrm{\u25b3}ABC$, but how?

Can anyone explain me this?

Properties of parallelograms
Answered

Arraryeldergox2
2022-06-27

Euclidean Geometry
Answered

Yahir Crane
2022-06-27

Euclidean Geometry
Answered

mravinjakag
2022-06-27

Postulates
Answered

Jeramiah Campos
2022-06-26

$\pi (2n)-\pi (n)\ge 1,$

for any $n>1$. The assertion we would like to prove is that the number of primes between $n$ and $2n$ tends to $\mathrm{\infty}$, if $n\to \mathrm{\infty}$, that is,

$\underset{n\to \mathrm{\infty}}{lim}\pi (2n)-\pi (n)=\mathrm{\infty}.$

Do you see an elegant proof?

Properties of parallelograms
Answered

flightwingsd2
2022-06-26

Here's the problem statement:

The area of the parallelogram with vertices 0, $\u043c$, $w$, and $v+w$ is 34. Find the area of the parallelogram with vertices 0, $Av$, $Aw$, and $Av+Aw$, where

$A=\left(\begin{array}{cc}3& -5\\ -1& -3\end{array}\right).$

I got the answer by doing something very tedious. I set $v={\textstyle (}\genfrac{}{}{0ex}{}{17}{0}{\textstyle )}$ and $w={\textstyle (}\genfrac{}{}{0ex}{}{0}{2}{\textstyle )}$, and did some really crazy matrix multiplication and a lot of plotting points of GeoGebra to get the answer of: $\overline{){\displaystyle 476}}$.

Now, I'm 100% sure that was not the fastest way, can someone tell me the non-bash way to do the problem?

Angle Bisectors
Answered

Gybrisysmemiau7
2022-06-26

Angle Bisectors
Answered

anginih86
2022-06-26

I cannot visualize this problem... If I draw a triangle and bisect the exterior angles, they never meet at a common point. Is this some sort of typo?

Midpoint Formula
Answered

Bailee Short
2022-06-25

Postulates
Answered

Extrakt04
2022-06-25

I know I have to Show that the parallel postulate 5 implies lorenz, and then lorenz implies parallel postulate 5.

Assume postulate 5 . So we are given AB and a point C not on AB. Choose B on AB draw CD to construct angle ECD= angle BDC.

I just don't get what Lorenz postulate means. Thats where I am getting stuck.

While elementary geometry is often encountered by college students as they are dealing with the basic design and modeling tasks, it is often necessary to implement equations or turn to Euclidean Geometry to determine the values of shapes and figures that are related to axioms and different theorems. When you are dealing with flat surfaces, it will help you to work with straight line segments. The same relates to those cases when you need to calculate formulas by using midpoint formula example problems that you can find below as you are dealing with the elementary problems in Geometry. If you find it ov