# Elementary geometry questions and answers

Recent questions in Elementary geometry
Waldronjw 2022-06-30

### Find the Midpoint Between the Given Points:$\left(1,1\right),\left(4,5\right)$

Banguizb 2022-06-30

### Let $\mathrm{△}ABC$ and E, D on $\left[AB\right]$ and $\left[AC\right]$ s.t. BEDC is inscribable. Let $P\in \left[BD\right],Q\in \left[CE\right]$ , s.t. AEPC and ADQB are also inscribable. Show that $AP=AQ$ .

Reginald Delacruz 2022-06-30

### In triangle $\mathrm{\Delta }ABC$ $\measuredangle C={90}^{\circ }$. The angle bisectors of angles $\mathrm{\angle }A$ and $\mathrm{\angle }B$ cross at point $O$. The distance from point $d\left(O,\overline{AC}\right)=3\mathrm{cm}$ and $d\left(O,\overline{AB}\right)=15\mathrm{cm}$. Find the perimeter of triangle $\mathrm{\Delta }ABC$ in cm

gnatopoditw 2022-06-30

### How to rigorously prove that the diagonals of a parallelogram are never parallel? It is intuitively obvious, but since it is not an axiom, it is a proposition that needs to be proved. I would like to see a proof without using analytic geometry, but only the old methods of Euclidean synthetic geometry.

DIAMMIBENVERMk1 2022-06-29

### I am beginning real analysis and got stuck on the first page (Peano Postulates). It reads as follows, at least in my textbook.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 $s\left(n\right)=1$.b. The function s is injective.c. Let $G\subseteq \mathbb{N}$ be a set. Suppose that $1\in G$, and that $g\in G⇒s\left(g\right)\in G$. 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 $\left\{1,3,5,7,\dots \right\}$, or the powers of 5 $\left\{1,5,25,625\dots \right\}$, could be constructed (with a different $s\left(n\right)$, of course, since $s\left(n\right)$ is not defined in the postulates anyway). How do these properties uniquely identify the set of the natural numbers?

Lucille Cummings 2022-06-29

### Finding the coordinates of a point five units along the line perpendicular to a midpoint?

dourtuntellorvl 2022-06-29

### Recall that Bertrand's postulate states that for $n\ge 2$ there always exists a prime between $n$ and $2n$. Bertrand's postulate was proved by Chebyshev. Recall also that the harmonic series$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$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 $ϵ>0$, there exists an $N$ such that if $n>N$, then there exists a prime between $n$ and $\left(1+ϵ\right)n$.Question: Is this conjecture true? If it is true, is there an expression for $N$ as a function of $ϵ$?

landdenaw 2022-06-29

### Given stepsizes ${h}_{1}$ and ${h}_{2}$ , develop a numerical scheme to approximate ${f}^{\mathrm{\prime }\mathrm{\prime }}\left({x}_{0}\right)$ with function values $f\left({x}_{0}\right)$ , $f\left({x}_{0}+{h}_{1}\right)$ and $f\left({x}_{0}+{h}_{2}\right)$ . Under what conditions will your method not work?

Quintin Stafford 2022-06-27

### The midpoints of the sides of $\mathrm{△}ABC$ along with any of the vertices as the fourth point make a parallelogram of area equal to what? The answer is, obviously, $\frac{1}{2}\mathrm{area}\left(\mathrm{△}ABC\right)$In the method, I took $\mathrm{△}ABC$ 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 $◻DEFB$.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 $◻DEBF$ will be 1/4 of $\mathrm{△}ABC$, but how?Can anyone explain me this?

Arraryeldergox2 2022-06-27

### A circle is circumscribed by a parallelogram. prove by using tangent property that the parallelogram is a rhombus. I tried to prove that the adjacent sides of the parallelogram are equal but I lack idea to apply tangent property

Yahir Crane 2022-06-27

### If in a tetrahedron ABCD the heights are congruent and A is projected on the (BCD) plane in the orthocenter, ABCD is a regular tetrahedron

mravinjakag 2022-06-27

### Prove: If the sum of the angles of a triangle is a constant n, then $n=180$ and thus the geometry is Euclidean

Jeramiah Campos 2022-06-26

### By Bertrand's postulate, we know that there exists at least one prime number between $n$ and $2n$ for any $n>1$. In other words, we have$\pi \left(2n\right)-\pi \left(n\right)\ge 1,$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,$\underset{n\to \mathrm{\infty }}{lim}\pi \left(2n\right)-\pi \left(n\right)=\mathrm{\infty }.$Do you see an elegant proof?

flightwingsd2 2022-06-26

### I had a question about a problem that I was working on for my pre-calculus class.Here's the problem statement:The area of the parallelogram with vertices 0, $м$, $w$, 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 $v=\left(\genfrac{}{}{0}{}{17}{0}\right)$ and $w=\left(\genfrac{}{}{0}{}{0}{2}\right)$, and did some really crazy matrix multiplication and a lot of plotting points of GeoGebra to get the answer of: $\overline{)476}$.Now, I'm 100% sure that was not the fastest way, can someone tell me the non-bash way to do the problem?

Gybrisysmemiau7 2022-06-26

### If I have triangle ABC with side lengths a,b,c, and I have an angle bisector coming out of point A, which divides side a into two sections at intersection point P, what is the length of BP and PC?

anginih86 2022-06-26

### Exterior angle bisectors of the side $\mathrm{△}ABC$ at vertices $B$ and $C$ intersect at $D$. Find $\mathrm{\angle }BDC$ if $\mathrm{\angle }BAC={40}^{\circ }$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?

Bailee Short 2022-06-25

### If the points $A\left(4,3\right)$ and $B\left(x,5\right)$ are on the circle with center $O\left(2,3\right)$ find the value of x.

Extrakt04 2022-06-25