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Questions tagged [geometry]

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

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Two definitions of $\pi$

I have the feeling that ancient mathematicians (like Greek or Chinese), trying to find good approximations of $\pi$ used two definitions: If $A$ is the area of a disk and $r$ is its radius, $\pi=A/r^...
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1answer
19 views

Find ratio of the volume of two cone

Given two sector ABC and PQR, $\angle A=2\theta$, $\angle P=3\theta, AC=2r, PR=3r, $ both sectors are folded into a right circular cone, find the ratio of the volume of two cone. I am having ...
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2answers
75 views

How to use an iterative method to compute y values corresponding to an x value of a rotated ellipse?

I am trying to render the outline of a rotated ellipse on raster graphics. The general form of a conic is: H(x, y) = A x² + B xy + C y² + D x + E y + F = 0 I am currently able to render the ellipse ...
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2answers
23 views

Formula (how to calculate) Y axis cross-point of two intersecting lines

i.e. I have two lines: A) Orange (Y axis starts at: 6, end at: -3) B) Green (...
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30 views

Discrete Geometry Pre-requisites

I have started studying Lectures on Discrete Geometry by Jiri Matousek, chapter 1 was fine for me I am midway chapter 2 but I am not understanding most of things after this. Are there any other books ...
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11 views

Parametric and implicit equations of $k$-flats (lines, planes, etc)

We know that any $k$-flat in $\mathbb{R}^n$, $k<n$, can be expressed either as a system of $m$ equations with $k=n-m$ and $$0 = c_{i,0}+\sum_{1\leq j\leq n} c_{i,j} x_j \quad ,i=1\dots m$$ or as $n$...
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40 views

Perspective Puzzlery! Determine Distance and Angle of View Point, from a Rectangle, in a Digital Image.

Given points; A - (341, 81) B - (265, 630) C - (884, 459) D - (896, 942) represent the coordinates of the corners of a rectangle in a digital photo. The true distance of Line AB=CD is 24 ...
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In $\triangle ABC$, if $3a=b+c$, then what is $\cos\frac{B}{2}\cdot\cot\frac{C}{2}$?

In $\triangle ABC$, we have that $3a=b+c$. Then, what's the value of $$\cos\frac{B}{2}\cdot\cot\frac{C}{2}$$ First I used auxiliary formulae of $\cos B/2$ with $3a=b+c\implies s=2a$, but I am ...
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1answer
28 views

How to find what degree on a circle is tangent to a point outside of that circle?

I know the (x,y) of a point, P, outside of a circle. I know the (x,y) for the origin of a circle, O. I know the radius, r, of that circle. How would I find what degree (e.g. 20 degrees, 270 degrees) ...
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39 views

Changing rotation center

Things that we have: 2 dimensions, a object with it's coordinates (object P1), it's rotation center (pivot) C1. After that lets rotate it at pivot C1 by known angle A. Now let's move that pivot by ...
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2answers
100 views

Proof that there exist only 17 Wallpaper Groups (Tilings of the plane)

I had a professor who once introduced us to Wallpaper Groups. There are many references that exist to understand what they are (example Wiki, Wallpaper group). The punchline is $$There \,\, are \,...
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1answer
108 views

For $P$ an arbitrary point in $\triangle ABC$, show that $\sum_{cyc}c(\sin \angle CAP+\sin\angle CBP)\leq a+b+c$

In the interior of $\triangle ABC$ we take the arbitrary point $P$. Prove that the following inequality holds: $$\small c(\sin\angle CAP + \sin\angle CBP) + a(\sin\angle ABP +\sin\angle ACP) + b(\sin\...
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11 views

Distance to median vs average intra-distances

Consider $n$ points in a vector space, denoted $(a_1, \dotsc, a_n)$. I am wondering if the following inequality holds true: $$ \min_x \sum_{i=1}^n d(a_i, x) \leq \frac{1}{n-1} \sum_{i=1}^n \sum_{j \...
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15 views

Computational geometry relationship between 2 arcs

I'm writing a program which is making offsets for provided shapes. On the attached picture you can see example of my arc object and all known values. Let's assume that a direction is CW. $O$ - ...
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1answer
139 views

Case for Pick Theorem? [on hold]

Could someone show or point me to a proof of the theorem for primitive triangles.
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1answer
47 views

Square with the minimum possible area that satisfy the given conditions

We're given a square with integer side lenght, it is divided into 4 triangles such that all the triangle's side lenghts are integers and that no two triangles are congruent, what is the minimum ...
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2answers
56 views

Applying Pick's Theorem

Let's say that $X$ is a parallelogram with vertices that have integer coordinates, how could I prove that $X$'s area is an integer? The vertices are $0, A, B$ and $A + B$. How would I do this?
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2answers
51 views

Area of triangle in $\mathbb R^3$ given $3$ coordinates

I know that if we have $3$ points $a$, $b$,and $c$ in $\mathbb R^3$, the area of the triangle is given by: $\frac{1}{2}\|\vec{ab}\times \vec{ac}\|$. This means that the area of the triangle equals ...
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1answer
22 views

Calculating a point in 3D using the Pythagorean theorem

I'm trying to find out if it's possible to calculate Az If I'm just given Ax and Ay. As an example say Ax is 3.4 and Ay is 7.8 how can I go about finding Az? If your wondering I was watching a video ...
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Proof that $n$ planes cut a solid torus into a maximum of $\frac16(n^3+3n^2+8n)$ pieces

Question: How many pieces a solid torus be cut into with three (affine) planar cuts? A google search will quickly reveal that the answer is thirteen, as can be read about here. The picture below ...
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34 views

Solving a area problem in a different way gives different solutions

today I was confronted with this little area problem from this video: https://www.youtube.com/watch?v=BgrWHOocYZA So I gave it a try and came up with a solution. But after comparing the solution with ...
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1answer
41 views

What is the radius of the semicircle?

Diagram : a 40cm wire bent to make a closed figure that consists a rectangle and a semicircle. (Kinda like a door with a curved top) Total perimeter of diagram / length of bent wire (rectangle + ...
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1answer
18 views

value of $m$ and $c$ in Circles

If line $y=mx+c$ is a common tangents to the given circles and $r_{4}=r_{1}+r_{2}$ and $r_{5}=r_{2}+r_{3}$ where $r_{i}$ is the radius of circle $C_{i}$ for $i=1,2,3,4,5$ Then value of $m+c$ is ...
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2answers
22 views

locus of point of intersection of family of lines

Consider the family of lines $(x-y-6)+\lambda(2x+y+3)=0$ and $(x+2y+4)+\mu(3x-2y-4)=0.$ If the lines of these two families are at right angle to each other. Then find the locus of point of ...
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1answer
32 views

Why are lengths different when an object is curved?

In the above diagram ad is equal to bc. Likewise, ce should be equal to df (Since angle F and E are at 90 degree). My question is why when calculating length of ce and df using $2\pi r$ they are ...
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0answers
41 views

When do three disks in the plane intersect?

Suppose $ABC$ is a triangle with $|AB|=c$, $|BC|=a$, $|CA|=b$. Suppose further that $A,B,C$ are the centers of three disks with radii $r_A,r_B,r_C$, respectively. Is there a sensible algebraic ...
3
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1answer
44 views

Characterizing triangles with integer-degree angles that are similar to any of their iterated orthic triangles

A triangle has integer angles $a,b,c$ (in degrees) and we create the triangle's pedal triangle and call it Pedal-$(1)$. (Here, "pedal triangle" means the specific pedal triangle whose vertices are the ...
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1answer
53 views

Using Ptolemy's Theorem to find length ratio

In this figure, $X, Y$ are tangent points and $\frac{DX}{EX} = \frac{8}{3} , \frac{EY}{DY} = 4 , \frac{AC}{AB} = \frac{5 }{4} . $ Then, what is $ \frac{BC}{AX}$ ? System of equations from the ...
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1answer
78 views

Is $[0,1) \cup \{2\}$ a manifold with boundary? My issue is the $2$.

This has been asked about here: Understanding topological and manifold boundaries on the real line, and Sharkos said Personally I'd say $M$ wasn't a valid manifold with boundary because the $\{2\}$ ...
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2answers
79 views

Foundational Areas of Math?

I am currently an undergrad student and recently changed my major to mathematics (after taking my first proof based math course). Unfortunately, now that I am taking several upper division math ...
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1answer
26 views

Difference between Orthic triangle and Pedal triangle

Is there any difference between Orthic triangle and Pedal triangle or both both are same ?
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0answers
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Does classification of 1-manifolds with boundary give induced orientation of image of closed interval under a smooth immersion?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. I have been trying to wrap my head around this for about 2 hours (...
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84 views

Using Coordinate Descent on Projected Space

My goal is to maximize an objective function using coordinate descent over a 3-dimensional vector. In the simple case the domain over which I am maximizing is defined as follows: $X \in \mathcal{X}$ ...
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2answers
44 views

Is there a closed-form solution for tangent circle in lens of two other circles?

I am given real values $p, s, t, u$ and wish to find unknown values $r, v$. As shown in the diagram below, $p$ and $s$ are radii of two given circles, with centers at $(0,-p)$ and $(0,t)$. At ...
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3answers
58 views

Existence of the inscribed hypersphere of a simplex

Letting $\textbf{T}=[u_0,...,u_n]$ be a $n$-simplex of $\mathbb{R}^n$, how does one prove the existence of the inscribed hypersphere ? Looking at the possible duplicates, people only seem to ask what ...
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2answers
51 views

In a quadrilateral ABCD ,which is not a parallelogramm. On the rays AB,CB,CD,AD we put… [closed]

In a quadrilateral $ABCD$, which is not a parallelogram, on rays $AB$, $CB$, $CD$, $AD$ we put points $K$, $L$, $M$, $N$ such that $KL\parallel MN\parallel AC$ and $LM\parallel KN\parallel BD$. Prove ...
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23 views

Circle can always represented as $\{z:|z-c|=k|z-d|\}$

I am reading Howie's Complex Analysis. There I see this remark: The observation that $c$ and $d$ are inverse points is the key to showing that every circle can be represented as $\{z:|z-c|=k|z-d|\}$...
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1answer
41 views

Plane distance from origin [closed]

Let the plane $V$ be defined by $ax+by+cz+d=0$ with $(a,b,c)$ not equal to $(0,0,0)$. Show that the distance between $V$ and origin is $\frac{d}{\sqrt{a^2+b^2+c^2}}$.prove the statement.
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1answer
8 views

can a 3D space be parallel/perpendicular to a plane or a line or another 3D space

As title. When I thought about parallel and perpendicular between line and line, plane and plane, line and plane, I want to go a step further find a 3D space having the relationships with another 3D ...
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1answer
33 views

The definition of the tangent vector of a manifold

In many textbooks of differential manifold, authors usually define the tangent vector on a manifold as below: Definition: Suppose $m$ is a $n$-dimensional smooth manifold, $x\in M$, the tangent ...
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1answer
43 views

$n$-gons gluing diagram/proof?

I am confused about this how I would visualize/prove this. Given that we have a regular $n$-gon that preserves orientation, what would the resulting surface be?
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2answers
40 views

Optimization path avoiding a set of points

I have a random polygon, convex or non-convex, defined by its vertices and two random points outside of the polygon (A and B) all of them defined in ${\rm I\!R}^{2}$, how can I get an optimized path, ...
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1answer
21 views

Why rotation of dodecahedron corresponds to an even permutation of inscribed five tetrahedra?

I’m reading Arnold’s Abel’s Theorem in Problems and Solutions, where it says: We now prove that the alternating group $A_5$ is not soluble. One of the possible proofs uses the following ...
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0answers
34 views

What's the name of the surface resulting from rotating the exponential curve?

In three dimensions, if we rotate the curve $(x,0,e^x)$ for $x\in\mathbb{R}$ around the axis through the points $(0,0,1)$ and $(1,0,0)$ by the angle $2\pi$, what's the resulting surface called? It ...
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1answer
41 views

Can anyone help me to find what is the similarity between these two triangles Just check the picture)? [closed]

Can anybody show me how does the red triangle (bigger triangle) is similar to the green triangle (smaller triangle). I need to know how they are similar so that I can do a proportional between them.
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0answers
22 views

Consequences of the axioms of connection and order

How do I prove that: Theorem 4. If we have given any finite number of points situated upon a straight line, we can always arrange them in a sequence A, B, C, D, E,. . ., K so that B shall lie between ...
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2answers
62 views

Compute numerically the angle $x$ in the triangle without trigonometry

Be $\triangle CAB$ right in $A$ such that $AB=a$ and $\angle CBA = \alpha$. Extend $BC$ to $D$ such that $\angle CAD=2\alpha$ and $\angle ADC=x$. If $M$ is a point $\in BC$ such that $BM=MC$ and $MD=...
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2answers
120 views

Geometry high school math competition question

Let $ABC$ be an equilateral triangle with side length $2$. Let the circle with diameter AB be $\gamma$. Consider the two tangents from $C$ to $\gamma$, and let the tangency point closer to $A$ be $...
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28 views

Given a set of points $S$ in the plane $\mathbb{C}$, is there a way to define “the smallest Euclidean shape that $S$ makes”?

For example, given $S := \{(0, 0), (0, 1), (1, -1), (-1, -1)\}$, I want to say, precisely, that the "smallest shape that $S$ makes" is something like the Star Trek symbol (https://1000logos.net/star-...
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27 views

Kind of hard(?) geometry proof [duplicate]

I am struggling with this proof. Consider the circle $\Gamma$ with a diameter $AB$. Another circle $\Lambda$ is drawn such that it is internally tangent to the $\Gamma$ and tangent to the diameter $AB$...