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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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0answers
26 views

how to find if it is possible to form a convex polygon with given n sides length? [duplicate]

I have given n lengths. I have to decide whether it is possible to construct an n-sided polygon with those n lengths. What are the necessary conditions or formula to check it?
3
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0answers
47 views

Numberphile's “plastic number” video, question regarding “calipers” used

Sorry for the poor title, but this question is awkward. Numberphile has a new video about the plastic number, and it demonstrates it with a set of calipers with 4 prongs. https://youtu.be/...
3
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4answers
241 views

Equilateral triangle on a concentric circle

Is my idea correct? 3 concentric circles of radius 1, 2 and 3 are given. An equilateral triangle is formed having its vertices lie on the side of the three concentric circles. What is the length of ...
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3answers
45 views

What's the size of the angle? [on hold]

Suppose we have a square ABCD. Point E is placed in the square so that DEC is an equilateral triangle. On the line BC, there is the point F, while $|EB| = |EF|$. What is the size of the $\angle CEF$?
2
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3answers
50 views

Some questions on the intersection of three cones.

I have three cones in $\mathbb{R}^3$, explicitly defined by the equations: $$ (x-\alpha_x)^2+(y-\alpha_y)^2=(z-r_1)^2 \,, \\ (x-\beta_x)^2+(y-\beta_y)^2=(z-r_2)^2 \,, \\ (x-\gamma_x)^2+(y-\gamma_y)^2=(...
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0answers
23 views

Quadric surfaces as loci of points

It is fairly usual to define the cuadratic sections as loci of points. For instance: Ellipse is a set (locus) of points $M$ the sum of whose distances to $F_1$ and $F_2$ is constant; Hyperbola is a ...
2
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1answer
37 views

Tangents of the Means of the Roots of $n$th Order Polynomials

In Stewart's Calculus w. Early Transcendentals 8th Ed. Chapter $3$ Problems Plus Question 26 we establish that the tangent of the mean of any two roots of a third order polynomial passes through the ...
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0answers
18 views

Section formula ratio assumption and confusion

What is the equation of the line which passes through $(-4,3)$ and the portion of the line intercepted between the axis is divided internally in the ratio $5:3$ by this point? This is easy but I am ...
2
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2answers
45 views

Volume of a spherical cap, where h < r [duplicate]

The question tells me to find the volume of a spherical cap with height $h$ and radius $r$ (by radius I mean radius of the cap, not the whole sphere). Well I attempted to solve it, as $R$ is the ...
0
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0answers
33 views

time to turn of 90° knowing speed and rotation angle [on hold]

im very bad in maths, but for some computer project i would need to get a formula allowing me to calculate the time it will take to turn of 90° if you know the speed (m/s) and rotation angle. to be ...
0
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2answers
39 views

Circle containing a point of square and touching two sides

So the problem goes,circle contains one vertex of a square and touches two sides. Length of side is 1cm. What is circumference of a circle? My attempt was trying to find a connection between radius of ...
0
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1answer
30 views

Quadrature of square's diagonal

I came up with this problem during shortening my road to an university on a car park. We will build a sequence of geometrical figures. As sequence's first element let's consider a square with a side ...
0
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1answer
39 views

ABCD and AECF are two parallelograms and side EF is parallel to AD . suppose AF and DE met at X and BF AND CE AT Y . prove that XY is parallel to AB

I tried proving it by showing angles exy and eyx equal to edc and ecd respectively but I got no where . Is there any other approach I should consider
0
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0answers
29 views

Pyramid and cone [on hold]

This is the first time I am seeking a question here. I need help on solving mathematics problem which I found in a local textbook for year 8. Though looks simple, somehow it gets too complicated and ...
0
votes
1answer
49 views

Prove that the straight lines whose direction cosines are given by the relations

Prove that the straight lines whose direction cosines are given by the relations $al+bm+cn=0$ and $fmn+gnl+hlm=0$ are perpendicular if $\dfrac {f}{a} +\dfrac {g}{b} + \dfrac {h}{c}=0$ and parallel if $...
0
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1answer
25 views

Rotation problem [duplicate]

With a 2D surface, we take $(2, 1)$ as the center point and consider a transformation with a rotation angle of $45^\circ$ so point $(3, 3)$ is transformed into point? I'm really close to getting the ...
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1answer
43 views

Find the angle c in a triangle abc. [on hold]

In a triangle abc , AD is the angle bisector of angle bac and meets BC at point D.angle bac is 111 degree.find angle acb. Also AB + BD = AC.
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0answers
18 views

Prove that a segment drawn from one vertex of a triangle to the opposite side is smaller than at least one of the adjacent sides of the triangle

Prove that a segment drawn from the vertex of a triangle to the opposite side is smaller than at least one of the adjacent sides of the triangle
0
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0answers
28 views

A monotone function must have a tangent ray that does not cross with the function

Consider a monotonically increasing and differentiable function $y=f(x)$ that passes through the origin. $\gamma=\{(x,y)|y=f(x)\}$ is the graph of $f$. Claim: there exists a ray $R$ such that $R\cap ...
2
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0answers
45 views

Explicit parametrization for a 3-ellipse? A 4-ellipse?

I searched around but was unable to find anything. For the usual $2$-ellipse we have the parametrization $x(t) = a\cos(t)$ and $y(t) = b\sin(t)$ for $t\in [0,2\pi]$. Is there anything similar for ...
0
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1answer
26 views

Angle of a Star Inscribed in a Circle

I don't even know where to start on this: In the figure, point O is the center of the circle, points A, B, C, D and E all lie on the circle, and both segment AD and CE go through point O. Angle BEC ...
3
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3answers
82 views

A monotonic function that intersect with all lines in $\mathbb R^2$

Let $f:\mathbb R\to\mathbb R$ be a monotone function. Let $\gamma=\{(x,y)\ |\ y=f(x)\}$ is a curve in $\mathbb R^2$. Does there exists a $f$ such that $\gamma\cap L\neq \emptyset \ \forall L\...
0
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1answer
40 views

Why can the gaussian curvature be computed this way?

I understood everything up to the last part ("It follows that the gaussian curvature $K = K(s, v)$ of the tube is given by..."). Why? What am I missing here? I know we can just compute it from the ...
1
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2answers
56 views

Area of the intersection of two triangles.

Let $\triangle{ABC}$ be a triangle with $AB=5$, $BC=7$, and $CA=4$. Define $D$, $E$, and $F$, to be the midpoints of $AB$, $BC$, and $CA$ respectively. Let $G$ the intersection of the medians of $\...
0
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0answers
19 views

Algorithm to arrange different-sized circles in a square area?

Suppose I have a large square and a set of $n$ circles, each with a different radius $r$, such that there exists some way to fit all the circles into the square. Is there an algorithm to find the "...
2
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0answers
31 views

How do I pack 2D objects into an arbitrary shape?

I've heard of packing problems in general, and I've seen a couple questions on this site that address specific cases (e.g. how to pack 45-45-90 triangles into an arbitrary shape), but after a search I ...
0
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0answers
38 views

Equilateral triangle. [on hold]

Let triangle $ABC$ is equilateral triangle. We choose any point inside this triangle. And we know the values of two angles $\alpha$ and $\beta$. What are the values of the angles in the triangle, ...
1
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1answer
25 views

Show that the set of lines in $\mathbb{R}^n$ is a (smooth) manifold of dimension $2(n-1)$

I was recently made aware of the result in the title. It's easy to show for $\mathbb{R}^2$, but I'm having trouble coming up with a generalization for $\mathbb{R}^n$. There are a couple of ways to ...
5
votes
1answer
50 views

What is the average distance of point in hypercube to its center?

How do I compute the average distance of point inside an hypercube to the center of the hypercube as a function of the dimensionality of the space? Here I consider the hypercube defined as $C_n=\{x\...
0
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1answer
17 views

Convert point from a plane to another

I have two planes: Plane A $[x,y]$ Plane B $[x,y]$ $[0,0]$ of $A$ is on $[0,0]$ of $B$. However, the axis $x$ of plan A doesn't have the same angle than the one of B. I have the angle of B, and the ...
3
votes
1answer
48 views

Eccentricity of conic given by a complicated equation with trigonometric coefficients such as $\tan 10^\circ$

Find the eccentricity of the conic given by: $$\left(x\tan 10^\circ+y\tan 20^\circ+\tan 30^\circ\right)\left(x\tan 120^\circ+y\tan 220^\circ+\tan 320^\circ\right)+2018=0$$ What I have tried $$\...
2
votes
2answers
78 views

In $\Delta$ABC, $AB=5$ and $E,F$ are two points on $BC$ such that $BE=1,EF=3,CF=2. $

In $\Delta$ABC, $AB=5$ and $E,F$ are two points on $BC$ such that $BE=1,EF=3,CF=2. AE$ and $AF$ intersect the circumcircle of $\Delta$$ABC$ at the point $G$ and $H$ respectively. $GH$ and $BC$ are ...
3
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3answers
667 views

How many of these lines lie entirely in the interior of the original cube? [on hold]

A portion of a wooden cube is sawed off at each vertex so that a small equilateral triangle is formed at each corner with vertices on the edges of the cube. The $24$ vertices of the new object are all ...
0
votes
1answer
58 views

value of $a+\frac{1}{b^2}$ in straight line

The sides of a triangle have the combined equation $x^2-3y^2-2xy+8y-4=0.$ The third side, which is variable always passes through the point $(-5,-1)$ . If the range of values of slope of the third ...
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0answers
90 views

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^...
0
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1answer
18 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 ...
0
votes
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 (...
0
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0answers
27 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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0answers
9 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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0answers
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 ...
2
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0answers
44 views

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
26 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) ...
0
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0answers
30 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 ...
6
votes
2answers
96 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
39 views

Special case for Giraud's Theorem

I was wondering how Giraud's Theorem would work for spherical polygons. Do we know a proof?
2
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1answer
96 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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0answers
53 views

$RBC$ and $RAB$ have the same measure [on hold]

Let $ABC$ be a triangle with right angle $A$. Let $AA' \perp BC$ and $C'$ the middle of $(AB)$. Let $P$ the middle of $AA'$, $Q$ the intersection of $CC'$ and $BP$, $R$ the ...
0
votes
0answers
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 \...
0
votes
0answers
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. $O$ - ...