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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Find radius of a circle using stewart theorm

A circle C of radius 5 cm and two circles C1 and C2 of radius 3,2 respectively . C1C2 touch each other externally and both touch C internally . A circle C3 touch C1,C2 externally and touch C ...
3
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1answer
39 views

Algebraic solution for the value of $x$.

I solved this problem the fifteen years ago without numerically solving equations of degree 4, I was happy in a substitution that I avoid directly attacking equations of degree 4. Today my nephew, ...
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32 views

Can anyone give me a solution with analytic geometry or complex Numbers?

The problem is a imo's problem. Triangle ABC has circumcircle H and circumcenter O. A circle R with center A intersects the segment BC at points D and E, such that B, D, E, and C are all different and ...
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2answers
51 views

Geometric interpretation of the geometric mean of two numbers

$a$ and $b$ are any two (positive) numbers. A geometric interpretation of the arithmetic mean and the harmonic mean of $a$ and $b$ are line segments parallel to the bases of a trapezoid of lengths $a$ ...
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1answer
33 views

a solid geometry problem

In the following 3D figure, we know that $AE \bot EC, AD \bot BD$, how to prove that $|ED| < |BC|$ ?
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2answers
60 views

What algebra is generated by $\mathrm{O}(2)$?

The unit complex numbers can be identified with the $2 \times 2$ special orthogonal matrices $\mathrm{SO}(2)$. The problem with $\mathrm{SO}(2)$, however, is that its not closed under $\mathbb{R}$-...
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2answers
37 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
28 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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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
56 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 ...
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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 ...
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28 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 ...
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1answer
27 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 ...
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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 ...
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1answer
57 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 \...
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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 : ...
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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 ...
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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 ...
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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 $AB$...
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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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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 $f(s)$ ...
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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 ...
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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 : $$(\...
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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((...
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2answers
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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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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 $(...
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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
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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, ...
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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 ...
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1answer
31 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 $...
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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 ...
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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 \...
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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
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2answers
46 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 ...
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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 ...
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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 ...
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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 $L$, $CD$ ...
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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 ...
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448 views

A space more fundamental than Euclidean space [closed]

Summary: The mathematical physicist Paolo Budinich attributes to Élie Cartan the statement that the geometry of pure spinors is "more elementary" or more "fundamental" than Euclidean geometry, which ...
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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'...
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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 ...
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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 $\overline{PO}$ (...
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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
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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 ...
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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
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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 $B$ ...
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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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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 ...