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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59 views

Let ABCD be a cyclic quadrilateral…

Let ABCD be a cyclic quadrilateral. Let r and s be the lines obtained by reflecting AB through the angle bisectors of $\angle CAD$ and $\angle CBD$, respectively. Let P be the intersection of r and s ...
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3answers
45 views

Let $T_1$ and $T_2$ be two circumferences…

Let $T_{1}$ and $T_{2}$ be two circumferences with centers $O_{1}$ and $O_{2}$ respectively, such that $T_{1}$ passes through $O_{2}.$ Let $C$ be a point on $T_1$. Let $r_{1}$ and $r_{2}$ be the lines ...
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689 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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1answer
21 views

Find a point further up on the steepest slope line in 3 dimensions?

I can find the coordinates of point B on the slope line in 2 dimensions. How do I find a similar point on the steepest tangent line in 3 dimensions? Starting with a point A with coordinates $\left(...
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3answers
77 views

Check if an ellipse is within an other ellipse

I have an ellipse $E_1$ centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane. How do I determine if another ellipse $E_2$ is within this given ...
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1answer
29 views

First and second fundamental form with rotational surfaces (check)

I'm working out some examples for surfaces in differential geometry. I was working out simple rotational surface, but I think I've done something wrong. Let $\gamma\left(t\right)$ a curve ...
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3answers
42 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
70 views

Geometric interpretation of the geometric mean of two numbers

$a$ and $b$ are any two (distinct, 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 ...
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1answer
113 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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2answers
51 views

Let ABCD be a parallelogram and let M be a point…

Let ABCD be a parallelogram and let M be a point on side AB. Let P be the intersection of side BC with the parallel line to AC that passes through M. Let Q be the intersection of AC with the parallel ...
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1answer
48 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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509 views

A space more fundamental than Euclidean space

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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1answer
38 views

Find radius of a circle using stewart theorem

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 ...
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1answer
38 views

intersection of three planes different cases, algebraic and geometric explanations

http://www.vitutor.com/geometry/space/three_planes.html Could anyone help me to understand the following cases $1.$ CASE (2.1), why rank of co-ef matrix is $2$ and augmented matrix is $3$? I can ...
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0answers
14 views

Is Invariance of domain an overkill?

Let $X=\Bbb R^{n}$ be the $n$-dimensional Euclidean space. Suppose that a map $f:X \to X$ is a contraction mapping, i.e., there exists $\delta \in [0,1)$ such that $$ |f(x)-f(y)| \leq \delta ...
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2answers
436 views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

I've thought about the following question for a month, but I'm facing difficulty. Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon ...
4
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2answers
66 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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1answer
379 views

A problem on triangle and its perpendicular bisectors.

I'm trying to solve the following problem : "In △ABC, coordinates of $B$ are $(−3, 3)$. Equation of the perpendicular bisector of side $AB$ is $2x + y − 7 = 0$. Equation of the perpendicular ...
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1answer
36 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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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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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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1answer
492 views

Do the tangents of two circles define concentric circles?

Given two non-overlapping circles, $R_1$ and $R_2$. The radii of $R_1$ and $R_2$ may be different. The distance between the centers of $R_1$ and $R_2$ is defined as $x$. Draw the four tangents ...
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1answer
1k views

How to plot $N$ points on the surface of a $D$-dimensional sphere roughly equidistant apart?

Let's say I have a $D$-dimensional sphere with a radius $R$. I want to plot $N$ number of points evenly distributed (equidistant apart from each other) on the surface of the sphere. It doesn't matter ...
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3answers
60 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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7 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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0answers
29 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
28 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
42 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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3answers
1k views

What happens when two infinite lines stop intersecting and become parallel?

I posited this question to my geometry teacher in highschool many years ago, and it stumped her. I've recently brought it up again in conversations with friends and have not gotten any answer that's ...
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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
1k views

calculating volume of a horizontal cylindrical tank from depth

Has anyone found a good formula to convert a depth of fluid to a volume remaining in a tank for a cylindrical tank laying horizontal? (with or without half sphere end caps) Much Thanks!
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1answer
17 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
16 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
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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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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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19 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
140 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 ...
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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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2answers
463 views

Properties of sphere

Let $C$ be a circle with diameter $\overline{AB}$. Then it is well known that for any $P$ on the circle $C$ the angle $\angle APB =\frac \pi 2$. There are similar results for sphere?
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800 views

Do collinear lines or overlapping collinear line segments intersect?

I am writing a function to find the intersection of a pair of lines and another function to find the intersection of a pair of line segments. The parallel case and the single point intersection case ...
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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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1answer
652 views

Find the shortest distance between $9x^2+9y^2-30y+16=0$ and $y^2=x^3$

Find the shortest distance between $9x^2+9y^2-30y+16=0$ and $y^2=x^3$. I know the shortest distance exists between the curves on the common normal line. Is there any other shorter way to attempt?
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28 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
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?
14
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1answer
288 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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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 ...