# Tagged Questions

For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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### Brownian Motion in Confined space, any results?

I am searching for work regarding Brownian motion in a confined space, like a sphere or a cylinder, where the wall will serve as reflection boundary. I am wondering if it is possible to derive results ...
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### Bending a line segment

REVISED QUESTION With the help of the existing answers I have been able to put together this clearer animation, and I asked this question to discover the shape is called a cochleoid. What I am ...
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### Connection of two objects in coordinate system

Please take a look at the following picture: I want to connect two objects by a line. This line has to start and end on an the red lines of the objects and are not allowed (at least it should be ...
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### In the given figure, $X$ and $Y$ are two centres

In the given figure, $X$ and $Y$ are two centres of two circles. They touch each other externally at a point $S$. $AB$ be the common tangent of both circles. $O$ be the centre of the third circle ...
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### How can I calculate the number of corners in a prism? [closed]

For any prism whose cross-section is a regular polygon of n sides, can it be shown that there is one corner for every m edges? If not, can an expression be found to calculate the corners?
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### Proving $\frac{1}{\cos^2\frac{\pi}{7}}+ \frac {1}{\cos^2\frac {2\pi}{7}}+\frac {1}{\cos^2\frac {3\pi}{7}} = 24$

Someone gave me the following problem, and using a calculator I managed to find the answer: $$\frac {1}{\cos^2\frac{\pi}{7}}+ \frac{1}{\cos^2\frac{2\pi}{7}}+\frac {1}{\cos^2\frac{3\pi}{7}} = 24$$ ...
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### A, B, C, D and E are five vertices of a cuboid. AE is a diagonal of the cuboid. Find the coordinates of E. [closed]

All coordinates are with respect to origin $O$. $A = (1,2,-1) B=(0,-3,-8) C = (4,-1,-10) D= (-4,2,-11)$ Find $E.$
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### Connecting $N \times N$ dots with straight lines

Many of you have likely seen the Connect 9 dots with 4 lines puzzle. My problem is similar, but more generic. Given an $N\times N$ square of points, what is the minimum number of lines needed to ...
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### Identify the given venn formula and describe its shape [closed]

Here is the image of the question.
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### How to scale a coordinate system?

I have a bunch of coordinates and I draw them on a Bitmap Image to then display them on an Image control in a Wpf application. Given data to take into consideration: My Bitmap File is ...
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### On ruler-only construction

There is a ruler with two different labels on it (1 inch, for example). The task is to find out if it is possible to construct a perpendicular to the given line. I have found a way to construct a ...
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### How to get neighbor points in high dimensional space?

I am writing to write an algorithm to do neighbor points search (so the implicit form will not work). We define neighbor points as the points with Euclidean distance $r$ to a given point. In 2D, for ...
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### Evaluate the angle x in terms of angles and y and z?

I need help with this problem and problems like this. I know that to solve this problem I have to find which angles are the same and etc, but how do I know that. How can I see that two angles are the ...
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### 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 ...
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### Loci of intersection points of two curves

There are two continuous, negatively-sloped curves,A and B. They intersect at least once ,say at $(x,y)$. If I introduce a third curve C, whose X axis intercept has a higher magnitude than that of B, ...
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### Clarification Needed:Equation Of Refracted Ray/Line

A ray of light is sent along the line $2x-3y=5$.After refracting across the line $x+y=1$ it enters the opposite side after turning by $15^0$ away from the line $x+y=1$.Find the equation of the ...
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### Determine the varying area of the shaded region

How to estimate the area of the shaded region shown in the attached Figure? Note that in the figure, $p$ has a maximum and minimum values of $p_a$ and $p_b$ respectively. Moreover, $p$ follows a ...
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### A rectangle of a given aspect ratio inscribed in a hexagon.

I'm trying to find the largest rectangle of a given aspect ratio that can be inscribed in a hexagon. I'm able to sort of walk through the problem in reverse, i.e. given an x, I can calculate the ...
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### On the relationship between $\text{SL}_2(5)$ and $A_5$ [duplicate]

I have two questions. What is the quickest way to see from scratch that $\text{SL}_2(5)/\{\pm I\}$ is isomorphic to the alternating group $A_5$? Does $\text{SL}_2(5)$ have any subgroups isomorphic ...
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### Finite subgroup of $\text{SO}(3)$ acts on set of points on unit sphere in $\mathbb{R}^3$ which are fixed via some nontrivial rotation in $G$

Let $G$ be a finite nontrivial subgroup of $\text{SO}(3)$. Let $X$ be the set of points on the unit sphere in $\mathbb{R}^3$ which are fixed by some nontrivial rotation in $G$. I have two questions. ...
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### Find the ratio of diagonals in Trapezoid

Given $ABCD$ a rectangular trapezoid, $\angle A=90^\circ$, $AB\parallel DC$, $2AB = CD$ and $AC \perp BD$. What is the value of $AC/BD$ ? Attempts so far: I have tried using the ratio of the areas ...