geometry assuming the parallel postulate of Euclid: in a plane, given a line and a point not on that line, there is exactly one line parallel to the given line through the given point.

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2
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2answers
29 views

Length of side of biggest square inscribed in a triangle

I have seen that the length of each side of the biggest square that can be inscribed in a right triangle is half the harmonic mean of the legs of the triangle. I have not seen a rigorous explanation ...
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2answers
26 views

Trying to find an isomorphism

I'm trying to find an isomorphism from Z31 to itself that maps 13 to 28.I used the extended euclid algorithm to find that 12 is the inverse of 13, but that's as far as I got. What do I do next? How ...
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0answers
26 views

Alternate form for the vector field $x^3\cdot\hat{x} + y^3\cdot\hat{y}$

Is there an alternate form for the vector field $x^3\cdot\hat{x} + y^3\cdot\hat{y}$ in which we can write all in function only of the radius $r=\sqrt{x^2 + y^2}$ ? Thank you
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3answers
54 views

How to find the area of the following isosceles triangle

I am stuck with the following problem : What is the area of an isosceles triangle whose equal sides are $20$ cm and the angle between them is $30^{\circ}$ ? It is a nineth standard problem and ...
0
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0answers
6 views

Fisheye equidistant projection mapping to fisheye stereographic projection?

I have a set of images captured by a wide-angle (fisheye) lens camera, and the projection is linear-scaled (equidistant). I would like to remap from this projection to fisheye stereographic, which is ...
0
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0answers
25 views

A generalization of the Sawayama lemma

Let $ABC$ be a triangle, let $D$ be a point on the line $BC$. The Thebault circle is a circle tangent $AD, BC$ and the circumcircle (yeallow circles in the following figure). I give a ...
1
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1answer
26 views

The intuition behind choosing this length?

In Problem 1, RMO 2004 there is a particular choice of length which leads to the solution, the length being that of the tangent from the foot of the perpendicular to the circle. Just a rundown of ...
0
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0answers
41 views

How much water would it take to fill a 1m^4 tesseract? Is it infinite? Do I need a 4D liquid?

Apologies, as I'm in no way a mathematically knowledgeable person. So this question may be proposed weirdly, or very simple. It's been evading my intuition for a while now, and I need a little help ...
2
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1answer
56 views

Question 5, RMO 2003, issue with ratios

In problem 5, RMO 2003 a specific part of the solution depends on the following $$\dfrac{BD}{DC} = \dfrac{AE}{EC} = \dfrac{AF}{FB} = \dfrac{DC}{BD}$$ It is proven that $AB \parallel DE \: , BC \...
5
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2answers
56 views

find point given a line and two arbitrary points on one side of the line

I have geometrical question which I'm trying to solve for a while now and it goes like this : ...
0
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1answer
15 views

How to describe this region in polar coordinates?

$D=\{(x,y)\in\mathbb{R}^2: x^2+y^2\leq 9 \text{ and } y\in [-3,1]\}$ I know how the region looks like but when $\theta \in [\sin^{-1}(\frac{1}{3}),\pi -\sin^{-1}(\frac{1}{3})]$ I don't know how to ...
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0answers
15 views

Linearity of projection of angle

In the book Putnam and Beyond, problem 252 reads as follows: Consider the angle formed by two half-lines in three-dimensional space. Prove that the average of the measure of the projection of the ...
0
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1answer
26 views

Weighted sum from Fermat point

Let $D$ be an interior point of triangle $ABC$ such that $\angle ADB=\angle BDC=\angle CDA=\frac{2\pi}{3}$. Find the minimum $k$ such that $k(AB+BC+CA)\geq 2AD+\frac32BD+CD$ is always true. The point ...
1
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1answer
32 views

Proving circumcenter lies on altitude

Problem: In $\triangle ABC$, let $D$ be the intersection of the tangents to the circumcircle at $B$ and $C$, let $B'$ be the reflection of $B$ across $AC$, let $C'$ be the reflection of $C$ across $AB$...
1
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1answer
20 views

Ray model in $3D$

Supposing we have a point source $p$ and multiple receiving points $r_i$ in $\Bbb R^3$ and there is a direct ray from $p$ to each $r_i$ and if there is a single mirror there is a single reflected ray ...
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0answers
18 views

How to shorten dot product

I would like to shorten a dot/scalar product: $$f(s)=sP_1+s^2P_2+\big((P_2-P_1)^TsN_1\big)N_1$$ Here $s$ are scalars, $P$ are points and $N$ are unit normal vectors in $R^3$. The function $f(s)$ ...
0
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1answer
45 views

Problem on Equilateral Triangle and points

Equilateral $\triangle{ABC}$ with sides $2\sqrt{3}$. Let $P$ be the point outside$\triangle{ABC}$ such that points $A$ and $P$ lie opposite to $BC$. Let $PD$, $PE$, $PF$ be the perpendicular dropped ...
0
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0answers
11 views

How to explain a sum of two mahalanobis projection?

I have to explain the use of the sum of two mahalanobis matrix the sum is done on the L component of the Mahalanobis matrix where $M=L^TL$ so i have $L=L_1+L_2$ and I formulated the following : $$(\...
0
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0answers
26 views

How to rewrite equation to get a quadratic patch

I would like to understand the given rewrite or transform from one equation to another. This is the original equation: $$p^*(q)=(u,v,w)\left( \matrix{q-n_i\big((q-x_i) \cdot n_i \big) \cr q-n_j \big((...
13
votes
2answers
1k views

What is the name of the circle that is tangent to three mutually-tangent circles centered at the vertices of a triangle?

I want some information about the little 'tangent circle', but I don't have its name to search for it in the internet. What is it called?
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0answers
33 views

12 points circle associated with a cyclic hexagon

When I research this problem A chain of six circles associated with a cyclic hexagon. I found the followings result. Let $ABCDEF$ be a cyclic hexagon. Let $A_1$ be any point on $AD$, the circle $(...
1
vote
1answer
26 views

Is there a simple way to decide if a hyperboloid is one-sheeted or two-sheeted, given the quadric equation?

Let us say that we have a quadric equation, whose solution set lies in $\mathbb{R}^3$, and you know it's a hyperboloid. Is there a way to analytically decide through a criterion if the hyperboloid is ...
14
votes
1answer
282 views

What's the best way to catch wild Pokémon in Pokémon GO?

In the newly released Pokémon GO, one of the major activities of the game is to catch wild Pokémon. These Pokémon are shown in the "nearby" list and their "rough distance" (RD) to you can be 0, 1, 2, ...
0
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0answers
30 views

How to define a transported version of the simplex $X = \{x \in \mathbb{R}^n_{+}| \sum\limits_{i = 1}^n x_i = 1\}$?

Let the simplex in $\mathbb{R}^n$ be denoted as $$X = \{x \in \mathbb{R}^n_{+}| \sum\limits_{i = 1}^n x_i = 1\}$$ So it looks like: I want to take a point $\bar x \in X$ (red dot) And drag it to ...
0
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1answer
30 views

Triangle Inequalities in Right Angled triangle.

In $\triangle{ABC}$, $\angle{ABC}=90^{\circ}$, $AB=BC$ and $AC=\sqrt{3}-1$. Suppose there exist a point $P_0$ in the plane of $\triangle{ABC}$ such that $AP_0+BP_0+CP_0 \leq AP+BP+CP$ for all points $...
0
votes
1answer
15 views

Three points and translation of the second

First of all, thanks for reading me and sorry for english mistakes. I have a programming--mathematical problem. Picture of the problem I have 3 coordonates in a 2d space, of 3 points. I want to ...
1
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1answer
32 views

The coincidence orthocenters of the two triangles

Let $CH -$ height in acute-angled triangle $ABC$. Some points $K$ and $N$ are on side $AB$. Let $O_1 -$ orthocenter of triangle $ACN$ and $O_2 -$ orthocenter of triangle $BCK$. Prove $$O_1=O_2=O \...
0
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0answers
35 views

Hippocrates trapezoid lune

How can I prove that a lune based on the construction of a constructible isosceles is quadrable? Hippocrates' other squarable lune
3
votes
2answers
45 views

Another formula for the angle bisector in a triangle

I have seen in an old geometry textbook that the formula for the length of the angle bisector at $A$ in $\triangle\mathit{ABC}$ is \begin{equation*} m_{a} = \sqrt{bc \left[1 - \left(\frac{a}{b + c}\...
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0answers
17 views

Applying delta method on euclidean distance

In order to estimate confidence interval of a k-dimension euclidean distance, I need to use delta method to estimate standard ...
0
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0answers
26 views

Number of neighbors as a function of dimension

I apologize in advance for perhaps an imprecise formulation of the question. If I have a point in 1D, it has precisely 2 nearest neighbors independent of choices. In 2D, if I allow arbitrary ...
0
votes
2answers
55 views

How to express an angle of 90 degrees between two lines?

If I would extend two lines $l_1$ and $l_2$ they would intersect with an angle of 90 degrees. How should I write with math terms that there would be a 90 degree angle. I assume $l_1 \perp l_2$ is ...
0
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0answers
22 views

A problem of understanding Pappus's theorem proof

I found a following theorem and its proof on a paper: Let $A, C$ and $E$ three points on a line and $B, D, F$ three points on another line. Let us suppose that $AB$ intersects $DE$ at point $L$, $CD$ ...
2
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0answers
45 views

Could Euclid have proven Dedekind's definition of real number multiplication?

In Euclid's day, the modern notion of real number did not exist; Euclid did not believe that the length of a line segment was a quantity measurable by number. But he did think it made sense to talk ...
1
vote
2answers
37 views

What do you advise me to do to understand middle school maths? [closed]

people. I am very new to stack exchange, and i just came into the maths section (i usually stick to coding section) because maths has started to get harder for me. I am now going to eight grade, and i'...
0
votes
2answers
53 views

Are there some undiscovered/unproved theorems about Euclidean triangles?

For a particular case of a figure called simplex, a triangle is surprisingly complicated (in my opinion). As an illustration, see the list of triangle topics on Wikipedia, and the Triangle page. The ...
0
votes
1answer
30 views

Does it have to be a right angle?

Say you have a circle $O$ and a point on the circle $P$. From P, we create 2 points $A$ and $B$ on the circle such that $PA=X$, $PB=Y$, and the 2 points are on different sides of $\overline{PO}$ (...
1
vote
1answer
45 views

Solving this without the concept of similarity?

Any point $X$ is taken on the side $BC$ of $\Delta ABC$. Prove that $AX$ is bisected by the straight line joining the midpoints of $AB$ and $AC$. This problem is trivial when one uses the concept of ...
2
votes
2answers
55 views

Is there a theorem of intersecting chords in an ellipse?

I found a well known theorem that if $A,B, C$ and $D$ are on the circumference of a circle and $AB\cap CD=P$ then $AP\cdot BP=CP\cdot DP$ . Is there anything generalization of it to an ellipse? Maybe ...
0
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0answers
25 views

pack equilateral triangle

I'm working on a problem of inscribing equilateral triangle for some time now and it goes like this : using only a foot rule and a compasses , show a way of inscribing an equilateral triangle into ...
6
votes
1answer
89 views

Proving an exercise from my High School Geometry Class

In my class we are learning geometry and the instructor gave us this problem: Let $ABC$ be a scalene triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ ...
2
votes
2answers
33 views

Prove on Incenter and mid point.

Let the incircle (with center $I$) of $\triangle{ABC}$ touch the side $BC$ at $X$, and let $A'$ be the midpoint of this side. Then prove that line $A'I$ (extended) bisects $AX$.
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votes
1answer
62 views

How do I convert $ 2=\sec(θ) $ into rectangular? [closed]

For my homework, I need to convert the polar equation of the "curve" $\sec\theta=2$ into rectangular form. I assume this must be an equation between $x$ and $y$, but the the answer according to my ...
0
votes
1answer
32 views

Intuitive explanation of Pascal's Theorem

I am wondering why Pascal's Theorem, as well as other 'Euclidean' results in projective geometry like Brianchon's Theorem should be true for not only circles, but also conics in general. Is there ...
0
votes
1answer
137 views

Would Euclid be satisfied by the construction of the 17-gon given by Gauss?

In our lecture on Algebra we were given the following exercice: Construct the regular 5-gon using straightedge and compass. (only using elementary geometric reasonig) If you construct the length ...
0
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3answers
47 views

What is the relationship between average distance between 2 random points in a cube, with min and max?

I wrote a computer program to simulate the minimum, average, and maximum distance between $2$ random points in a unit length cube (such as $1$ cubic foot). The minimum possible distance is $0$ and ...
2
votes
2answers
62 views

Solve using Butterfly Theorem.

Let $PT$ and $PB$ be two tangents to a circle, $AB$ the diameter through $B$, and $TH$ the perpendicular from $T$ to $AB$. Then prove that $AP$ bisects $TH$.
0
votes
1answer
18 views

Prove between Simson line & Nine point circle.

Prove that the Simson lines of diametrically opposite points on the circumcircle are perpendicular to each other and meet on the nine-point circle. I proved the first part of the problem but not able ...
0
votes
1answer
47 views

Circumcenter of triangle

I already know that, for a triangle $\Delta ABC$ $G$ is the triangle centroid, we have $$\vec{GA} + \vec{GB} + \vec{GC} = \vec{0}.$$ $I$ is the incenter, we have $$a\vec{IA} + b\vec{IB} + c\vec{IC} =...
2
votes
2answers
65 views

Euclidea 3 9.8 Chord Trisection

Construct a chord of the larger circle through the given point (on circumference of larger circle) that is divided into three equal segments by the smaller circle (circles are concentric) I'm having ...