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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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26 views

Line through a a given point in the first quadrant of the coordinate plane to form a triangle [closed]

Consider a straight line with negative gradient passing through the positive quadrant (where all co-ordinates are positive, or the first quadrant) of the co-ordinate plane and intercepting the $x$ and ...
2
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0answers
19 views

distribution of bisecting great circles

For any point on the globe, I believe there is (by the mean value theorem) at least one great circle containing that point and dividing the world's land area (or water mass, or population, whatever) ...
2
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0answers
80 views

Find Levi-Civita Connection in Hyperbolic Space

With: $$ \mathbb{H}^n=\left\{ (x_0,x_1,\dots, x_n)\in \mathbb{R}^{n+1}: \; x_0^2=1+x_1^2+\cdots +x_n^2,\; x_0>0\right\}. $$ and the form $$ \langle\langle(u_0,u_1,\dots, u_n),(v_0,v_1,\dots,v_n)\...
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0answers
19 views

Circular Clothoid Curve

One clothoid curve (see figure) has the curve parameters $(C^2= 9$ x $10^8 m^2, L= 315m)$. The starting point A of this curve is the beginning point of a road. The coordinates of point A are $(2000....
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0answers
22 views

Effect of plane isometry on punctured disc

I want to confirm if I am correct. Due to the rotational symmetry of open unit disc $D$, it suffices to study only translations on $D$ out of the four plane isometries? This is because, every other ...
2
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1answer
54 views

Geometrical problem in Newton's “Principia”.

Let VQPA be the circumference of the circle, S the given point toward which the force tends as to its center, P the body revolving in the circumference, Q the place to which it will move next, and PRZ ...
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2answers
24 views

Minimising the Sum of distance of two line formed by joining some point on a line to two fixed points

Consider the xy coordinate system. Let $l$ be the line $y=-x$. Consider two points $A=(0,2)$ and $B=(2,3)$ The Question asks to find a point $C$ on the line $l$ such that $|AC|+|BC|$ is minimum. ...
3
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1answer
23 views

Calculating the area of the parallelogramm given $4$ vertices.

I want to calculate the area of a parallelogramm given the following four vertices: $$\vec{p}=\begin{pmatrix}2 \\ 0\\3 \end {pmatrix},\vec{q}=\begin{pmatrix}8 \\ 1\\1 \end {pmatrix},\vec{r}=\begin{...
3
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0answers
95 views

a problem about the incircles of two triangles that the orthocenter formed.

See below diagram. $H$ is the orthocenter of an acute triangle $ABC$ where $AB \neq AC$. The circle centered at $I$ and the circle centered at $J$ are the incircles of triangles $ABH$ and $ACH$. $XY$ ...
3
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2answers
30 views

How many uniform polytopes are there in higher dimensions?

I am not really interested in the exact numbers, but more in the richness of the class of uniform (convex) polytopes in higher dimensions. Wikipedia contains the followin statement: In five and ...
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1answer
19 views

Proving that these four vectors form a parallelogramm

I am given four vectors in three-dimensional space and I want to check if they form a parallelogramm. The vectors are: $$\vec{p}=\begin{pmatrix}2 \\ 0\\3 \end {pmatrix},\vec{q}=\begin{pmatrix}8 \\ 1\\...
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0answers
20 views

How to create this specific geometric template?

I'm trying to find a way to create this geometric template... I simply need to find a way to create a circle with my desired number of points spread evenly around it.
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2answers
41 views

Chord of a Circle [closed]

Simple question regarding chords in a circle. Is the midpoint of a circle's chord always perpendicular to the circle's centre?
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1answer
45 views

How to prove that the midpoint of the given line segment lies on another line segment?

I have the following question with me: In a triangle ABC D and E lie on sides BC and AB respectively. F lies on AC such that EF is parallel to BC, G lies on side BC such that EG lies parallel to AD. ...
0
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1answer
28 views

Are 2D grayscale images actually 3D?

I keep running into this issue when describing the dimensionality of data I work with. In general it's a question of whether or not the 3D timeseries we are handling are in fact 5D, but I guess the ...
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3answers
72 views

What is the volume of the region $S =\{(x, y, z) : |x| + |y| + |z| ≤ 1\}$?

What is the volume of the region $S =\{(x, y, z) : |x| + |y| + |z| ≤ 1\}$ ? How can i find the volume any hints /solution
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2answers
37 views

Proving 2 chords are the same length in a circle divided into $n$ equal arcs

Let us have a circle that is divided in to $n$ equal arcs by $n$ points on the circumference. There are $\dfrac{n}{2}$ chords joining pairs of points. For what values of $n$ would it be necessary for ...
4
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1answer
38 views

Chord partition of regular polygon: same fraction of area and perimeter?

This is a variation of a question posed by James Tanton on Twitter. Let $P$ be a regular $n$-gon, $n \ge 3$. A chord $c$ of $P$ is a segment connecting two distinct points of the boundary of $P$, on ...
2
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1answer
77 views

Confused about rotation matrices

Applying a rotation matrix to a vector means shifting its coordinates to perform the rotation effect. Applying a rotation matrix to a model at the origin $(0,0,0)$ is not the same at performing a ...
3
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1answer
52 views

How can people assume this angle is exactly half of the other angle?

Long story short what I don't understand is underlined here in red: So, they somehow seem to assume the angle on the triangle on the right has an angle $\frac{\theta}{2}$. How do they know that? ...
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0answers
44 views

What software can draw all pictures for Trigonometric expressions like this?

I am preparing these pictures manually for Structural designs enter image description here I want some software which will automatically generate all these possible pictures from the given ...
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0answers
49 views

Prove the size of a hyperbolic angle is twice the area of its hyperbolic sector.

I'm trying to figure out how the hyperbolic functions are derived using a unit hyperbola. According to this walkthrough, argument u in (cosh(u), sinh(u)) should be equal to 2A, where A is the area ...
2
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2answers
46 views

In $\Delta ABC$ if $(\sqrt{3}-1)a=2b$, $A=3B$, then find $C$

In $\Delta ABC$ if $(\sqrt{3}-1)a=2b$, $A=3B$, then find $C$ My Attempt $$ b=\frac{\sqrt{3}-1}{2}a\quad\& \quad \frac{A-B}{2}=B\quad\&\quad\frac{A+B}{2}=2B\\\frac{a-b}{a+b}=\frac{\tan\frac{A-...
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2answers
73 views

Do the given perimeter and area corresponds to many shapes? [closed]

I have a perimeter P and area A of a planar shape. How to prove that there are many shapes that corresponds to those perimeter and area values?
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1answer
65 views

In $\triangle ABC$ with $D$ on $\overline{AC}$, if $\angle CBD=2\angle ABD$ and the circumcenter lies on $\overline{BC}$, then $AD/DC\neq 1/2$

Let $\alpha$ be $\measuredangle ABD$ Let $\beta$ be $\measuredangle DBC$ Let D be a point on AC such that BD passes through the origin point O Prove that $\frac{AD}{DC}$ cannot be equal to ...
2
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3answers
79 views

Euclidean geometry book for math contests

I'm a last year high school student, I'm looking for a "short" (by short I mean, not over 250 pages) Euclidean geometry book that covers topics linked to euclidean geometry of math contests, I have a ...
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2answers
35 views

In $\Delta ABC$, find $\cot\dfrac{B}{2}.\cot\dfrac{C}{2}$ if $b+c=3a$

If in a triangle ABC, $b+c=3a$, then $\cot\dfrac{B}{2}.\cot\dfrac{C}{2}$ is equal to ? My reference gives the solution $2$, but I have no clue of where to start ? My Attempt $$ \cot\dfrac{B}{2}.\cot\...
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1answer
63 views

Circles within ellipses

What is the largest circle centered at $(x_0, y_0)$ that is totally enveloped by an ellipse with a major axis $A$ and minor axis $B$? In this problem, assume a constraint s.t. the major axis is in the ...
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2answers
31 views

3D geometry, what are the coordinates of the 4th vertex and the point of intersection of this trapezoid?

3 Vertex of the trapezoid are given : A(4,-1,2) B(7,1,-3) D(0,-4,6) and we know that AB and CD are parallel, and CD=2AB (opposite vertices are B-D and A-C) The question is : what are the coordinates ...
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1answer
60 views

Show that $CD\perp AB $.

Let $\triangle ABC $ with all angles $<90°$. Let $A_1$ the middle of $ [BC] $. If $\angle BAA_1=30°$ and $D\in [AB] $ s.t. $CD=AB $ show that $CD\perp AB $. My idea : I draw $A_1T\perp AB $, $T\...
4
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1answer
74 views

Generalized AM-GM Inequality

I was discussing means with my friend, and I tried to illustrate the concept of geometric mean using the following idea: Suppose we have two positive quantities $x,y>0$. The simplest geometric ...
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1answer
15 views

Hausdorff measure and angle of a regular simplex/tetrahedron

Let $T \subset \mathbb{R^3}$ be a regular simplex/tetrahedron with boundary $\partial{T}$ and edge lenght 1. In which way can the angle between two different edges be computed? I know the formula ...
9
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4answers
257 views

Packing regular tetrahedra of edge length 1 with a vertex at the origin in a unit sphere

I'm currently venturing through Paul Sally's Fundamentals of Mathematical Analysis. This is an unusual textbook in terms of the difficulty of exercises. I've already been stunned by the very first one:...
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2answers
36 views

How does one explicitly show that the distance from Focus_1 to the ellipse plus the distance from Focus_2 to the ellipse is a constant?

When deriving the equation for an ellipse, I have seen that one of the first steps is to recognize that for a point on the ellipse defined by y=0, the two distances from Focus_1 (call it D1) and ...
1
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1answer
42 views

Typo? $H$ is quaternion algebra and $C$ is complex number. $M_n(H)\cong M_{2n}(C)$?

Denote $H$ as quaternion algebra and $C$ as complex number. Take any $a\in H$. One has $a=(a_1+ia_2)+(a_3+ia_4)j=c_1+jc_2$ with $ij=k$. Denote $M_n(F)$ as the matrix entries in division algebra $F$. ...
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0answers
25 views

Is there a name for the extent of an object in a 4th and higher spatial dimensions? (width, height, depth, …) [duplicate]

If width is the extent in the first dimension, height in the second and depth in the third, is there a name for the extent of an object in the fourth spatial dimension? And what about a five, sixth ...
4
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1answer
28 views

Cannot prove a geometry area ratio between a triangle and a parallelogram

ABCD is a parallelogram. Prove the following: $\frac{BF}{FA} = \frac{AD}{AE}$ $\frac{S_{ADF}}{S_{AEF}} = \frac{AD}{AE}$ $S_{EBF} = S_{ADF}$ $S_{BCE} = \frac{1}{2}S_{ABCD}$ I solved ...
2
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1answer
53 views

The Image of sphere under the map $f: (x,y,z) \to (x^2,y^2,z^2,\sqrt{2}yz, \sqrt{2}zx, \sqrt{2}xy)$ and $\mathbb{R}P^2$

I'm going to take a course about diferentiable manifolds next semester and I'm preparing for it by solving some of the problems from the book An Introduction to Differential Manifolds by D. Barden and ...
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0answers
31 views

Show that $AB=AA_1$

Let $ABC $ a triangle with $ A_1$ the middle of the edge $[BC] $ and $BAA_1=30°$. $ Let D\in [AB] $ s.t. $CD=AB $. I have to show that $AB=AA_1$. This conclusion seems to be wrong because I can ...
4
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1answer
57 views

Napoleon-like theorem concerning squares erected on sides of midpoint polygon of octogon

Given an arbitrary octagon, construct it's midpoint polygon(the midpoint formed by the midpoints of the sides). Erect squares on the sides of the midpoint polygon, all inwards or all outwards. ...
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2answers
38 views

Line Integral Harmonization

Is there a connection between line integrals over scalar fields and line integrals over vector fields? For example, do the pair $f(x, y)$ and $F(x, y)$ which stand in a potential function and ...
4
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0answers
87 views

How do we deduce that an ellipse, when defined as a “stretched circle”, has foci? [duplicate]

I've learned two separate ways to define an ellipse: It's a stretched circle. We get the formula for a unit circle, $X^2 + Y^2 = 1$, and stretch it by dividing the terms like so: $(\frac{X}{a})^2 + (\...
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1answer
27 views

Constant lengths (dimensional constants) and scaling

Suppose in the $xy$-plane we have defined the constant length $L$. This can be a fixed radius of a circle; or a boundary condition or any condition such, that $L$ has dimension of "meters" and is ...
4
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2answers
167 views

Prove triangle area formula for barycentric coordinates

Let $P_1, P_2, P_3$ be points with barycentric coordinates (with reference triangle $ABC$) $P_i = (u_i, v_i, w_i )$ for $i = 1, 2, 3$. Then the signed area of $\Delta P_1P_2P_3$ is given by the ...
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2answers
43 views

Prove that a circle can be inscribed iff the given condition is satisfied

I have the following question with me: "Let $B_1$ and $C_1$ be points on the sides $AC$ and $AB$ of a triangle $ABC$. Lines $BB_1$ and $CC_1$ intersect at point $D$. Prove that a circle can be ...
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1answer
66 views

how is the following curve not simple curve? [closed]

if we start from just above point and go up and end at just below point how is the curve not simple curve. ncert class 6 chapter 4 pg 72 says it is not http://ncert.nic.in/textbook/textbook.htm?femh1=...
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1answer
37 views

Minimum number of moves required to obtain chess like coloring.

I've been preparing for a competition and there is this problem that I cannot solve. Can you please help me and also tell me how to do similar problems if they appear in the future? Problem:
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votes
4answers
199 views

Defining the Cosine Function from First Principles, intuitively

Throughout most of my mathematics education, the cosine function has been defined formally either using its series expansion, as $\mathfrak{Re}(\exp i\theta)$, or as the unique solution to $y+y''=0$ ...
1
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1answer
36 views

Generalized Circumcenter: minimizing the range of distances from a point to the vertices of a polygon

It is well known that the circumcenter of a polygon exists if and only if the polygon is cyclic. I would like to extend the definition of a circumcenter for noncyclic polygons. Let us define $c(A)$ ...
4
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3answers
78 views

Can't figure out this triangle geometry problem

I have the following triangle: The following information about it are given: ABCD is a trapezoid (AB || DC) EF || DC Q is the intersection of AC, DB, PN, & EF Prove that EQ =...