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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3answers
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Visual Interpretation for the Sum of a Finite Geometric Series

I'm interested in intuitive visual explanations for the sum of a finite geometric series. I know there are some pretty "intuitive" explanations out there (including some on this site), but I haven't ...
2
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1answer
42 views

Failing to visualise Steiner's argument - Isoperimetric Theorem

My question is regarding Steiner's (incomplete) proof for the Isoperimetric problem, as presented in the book What is Mathematics. In a critical step, Steiner asks to readjust half the 'assumed' ...
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1answer
28 views

inequality for lengths of projections inside a circle

In the following picture, how can show that $c \leq a + b$? In the picture, $x$, $y$ and $z$ are three vectors of equal length. We can split $x$ in a component parallel to $y$ and a component ...
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0answers
22 views

Points on two perpendicular lines [on hold]

I got in troubles. I have no idea how to work with 3D lines, just with 2D lines. My problem is this: I got a point on a line and a point on a perpendicular line in 3D space. I have to get the gradient ...
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0answers
34 views

For non-negative $a$ and $b$ with $a+b \leq c$ for a small constant $c$, what is the minimum of $\cos a + \cos b$?

Let $a,b \geq 0$ with $a+b \leq c$ for a small constant $c$ between $0$ and $1$. What is the minimum of $\cos(a) + \cos(b)$? I conjecture it is $\cos(0)+\cos(c) = 1 + \cos(c)$ but I have no ...
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5answers
39 views

The horizontal side length of one of four small rectangle is a and the vertical side length is b. Verification: a + b = 1.

A square ABCD with a side length of 1 is divided into five small rectangles, four of which have the same area. The horizontal side length of one of four small rectangle is a and the vertical side ...
2
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4answers
51 views

Square with midpoints of $AD$ and $CD$

Square $ABCD$ is given; $MA=MD=ND=NC$. Show $AF=AB$. The first thing I noticed was $\triangle CDM \cong \triangle BCN$ and we obtain $CM = BN$ and $\angle MCD = \angle NBC$. Now I am trying to ...
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2answers
21 views

Finding the points that line on a plane

Let $P$ denote the plane given by the point-normal equation: $0 = (1,2,−1)·((x,y,z)−(1,1,1))$ How do I find the points $(x, y, z)$ that lie on the plane $P$?
2
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0answers
61 views

Compute face angles for a snub disphenoid (Johnson solid J84)

What I understand so far: This shape has 12 triangular faces (all equilateral triangles since this is a Johnson solid) and 8 vertices. We have 4 faces meeting at 4 of the vertices and 5 faces meeting ...
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0answers
26 views

Geometry of Envelope form definition

I had read about the envelope of the family of the curve. It is defined as a curve which is tangent to each member of the family at a single point and it is union of all such points. To find envelope ...
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0answers
16 views

Prove that orthocenter of an orthic triangle is the bigger triangles circumcenter [on hold]

I have tried to prove this fact for some time now.I tried angle chasing, looking at cyclic quadrilaterals and looking for similar triangles but none of it is taking.
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2answers
70 views

Geometry : what is the $\phi$ angle, if area of yellow rectangle is equal with area of red triangle?

I have a right triangle and in it area of yellow rectangle is equal with area of red triangle. How could prove that $\phi=45^{\circ}$? $$\text{Area of Yellow Rectangle}=\text{Area of Red Triangle}...
19
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5answers
3k views

Is there a known non-euclidean geometry where two concentric circles of different radii can intersect? (as in the novel “The Universe Between”)

From the 1951 novel The Universe Between by Alan E. Nourse. Bob Benedict is one of the few scientists able to make contact with the invisible, dangerous world of The Thresholders and return—sane! ...
0
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2answers
34 views

Can we define the apothem for any triangle?

Is there an apothem for any triangle, because every triangle can be circumscribed in the circumference, and so we have the radius of the circumference inscribed? Is there or is there not an apothem ...
2
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1answer
45 views

Generalizing the “The Volume of a Cone is a Third that of its Bounding Cylinder” fact

The ancient result is that a right-circular cone of height $h$ and base-radius $r$ will have volume $\frac{1}{3} \pi r^2h$, which is $1/3$ the volume of the cylinder with same base and height. And the ...
4
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1answer
48 views

Cyclic quadrilateral and trapezoid

A circle with diameter the minor base $CD$ of a trapezium $ABCD$ intersects its diagonals $AC$ and $BD$ in, respectively, their midpoints $M$ and $N$. The lines $DM$ and $CN$ intersect in $P$ and $AC$ ...
0
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1answer
38 views

Flat geometry, What is the value of BC Side?

In a triangle ABC have AB =10cm and AC=12cm. The incentro(I) and the baricenter(B) are in the same parallel to BC. The BC side measurement is equal to: I have developed so far: I did not calculate ...
1
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1answer
25 views

infinite order automorphism on torus

Let $T= \mathbb R^n/\Gamma$ a torus, where $\Gamma$ is the standard lattice in $\mathbb R^n$. A matrix $A\in SL(n,\mathbb Z)$ induces an automorphism $A_T$ on $T$. I want to calculate the order of $...
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0answers
14 views

Is the ratio of displacement to path length a reasonable measure of the curl of a path?

I am analyzing monte carlo generated data of the paths of some particles. For each event, the path is (nearly) random except that the probability of the angle of deflection at each interaction point ...
5
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1answer
68 views

Generalizing Archimedes' “The Quadrature of the Parabola”

In the third century BC Archimedes discovered that The area enclosed by a parabola and a line (left figure) is 4/3 that of a related inscribed triangle (right figure). Consequentially, the area ...
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2answers
49 views

Prove that $A, P, Q$ are collinear.

Two circles $\omega_1$, $\omega_2$ intersect at $A, B$. An arbitrary line through $B$ meets $\omega_1$, $\omega_2$ at $C, D$ respectively. The points $E, F$ are chosen on $\omega_1$, $\omega_2$ ...
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0answers
22 views

Reference Request: Algebraic and/or geometrical study on closed flat space

I am curious about studying the following subset $X$ of $\mathbb{R}^n$: $$X:=\{(x_1,\cdots,x_n)\in\mathbb{R}^n|x_i\ge 0\ \mathrm{for}\ \forall i,\sum_{i=1}^nx_i=1\}.$$ I can see that $X$ is ...
0
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2answers
37 views

ABCD is a rectangle with AB=8 inches and BC=6 inches. CE is drawn parallel to both CD and BC at C. If EC=4 inches, find the length of AE.

I know this is a fairly simple question, I got the answer to either be $2\sqrt{41}$ or $6\sqrt{5}$, however its a multiple choice and the answers are either $10.77, 9.17, 11.22,$ and $10$. I'm not ...
1
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2answers
42 views

Co-ordinate Geometry: Equation of the line

Find the equation of the line that passes through the point of intersection of the lines $3𝑥 + 2𝑦 − 1 = 0$ and $2𝑥 − 𝑦 + 7 = 0$ and is perpendicular to the line $4𝑦 + 4𝑥 = 7$. Solve for x: $3�...
2
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2answers
81 views

Limit of sum of areas of infinite amount of triangles

I apologize for the possible incorrect use of math terms since English is not my native language and I'm not a mathematician, but this issue came to my mind about a month ago and I was unable to solve ...
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2answers
30 views

A vector bisects the angle between two vectors

Find the vector C for which C bisects the angle between the two vectors A and B where $A = (2, -3 ,6) , B = (-1 , 2 ,-2)$ And the norm of C = $3\sqrt{42}$ My turn : Let $$ C = (x ,y,z) $$ then $$\frac{...
2
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4answers
73 views

Angles between vectors of center of two incircles

I have two two incircle between rectangle and two quadrilateral circlein. It's possible to determine exact value of $\phi,$ angles between vectors of center of two circles.
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1answer
28 views

Application of hyperspheres [on hold]

Recently I've been studying the the volume of an n-ball. Do hyperspheres (or their volume/surface formulas) have any real-world applications?
1
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1answer
28 views

Kiselev's Geometry: Find a point $C$ such that the sum $CA + CB$ is congruent to a given segment.

I am trying to solve exercise 139 from Kiselev's Geometry Book 1. Planimetry Given a point $A$ on one of the sides of an angle $B$. On the other side of the angle, find a point $C$ such that the sum $...
1
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1answer
29 views

A systematic way to find scaled/rotated hexagonal lattices commensurate with a unit hexagonal lattice?

Wood's notation is used to reference the relationship between two 2D lattices when the angle between the two unit vectors $(a_1, a_2)$ of one lattice is the same as the angle between the unit vectors $...
0
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0answers
21 views

Is it possible to make a stereographic projection of a stereographic projection?

Since we can't visualize a 3-sphere directily, I was wondering how does its stereographic projection look like, and found this neat demonstration: https://demonstrations.wolfram.com/...
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0answers
17 views

Divide circle and subcircle evenly by area

How do you divide a circle into a certain number of shapes with the same area while having at least one sub circle dividing the whole figure? I know that if we divide along the center of the circle in ...
0
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2answers
27 views

Calculating the coordinates of a unit vector normal

First of all I apologize if the question is elementary, I have not practiced math for a long time and have only just slowly picked up the fundamentals again. (i) With reference to Figure Q4, ...
1
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0answers
27 views

Elliptic element points of order 2 and 3 in the Fundamental Region for $PSL(2,\mathbb{Z})$

I am aware the Fundamental region for $PSL(2,\mathbb{Z})$ has 3 vertices, namely: $\rho = \frac{-1+i\sqrt{3}}{2}$, $\rho +1 = \frac{1+i\sqrt{3}}{2}$ and $i$, each stabilised by the cyclic subgroups ...
20
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4answers
3k views

Trying to visualize the hierarchy of mathematical spaces

I was inspired by this flowchart of mathematical sets and wanted to try and visualize it, since I internalize math best in that way. This is what I've come up with so far: Version 1 (old diagram) ...
0
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2answers
45 views

Flat geometry(what would the drawing of this figure look like?including the values)

Consider a right-angled triangle and the circumference inscribed on it. The point of contact between the hypotenuse and the circumference determines in the hypotenuse segments 4meters and 6meters, ...
1
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1answer
33 views

How to compute equally spaced points on involute curve

my problem is that I want to compute equally spaced points on involute curve. Starting from the beginning the involute curve parametric equations are: $$x(t) = r \cdot \sin(t) - r \cdot t \cdot \cos(...
2
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1answer
39 views

Every hyperbolic and parabolic cyclic subgroup of $PSL(2,\mathbb{R})$ is Fuchsian

I am working with the book "Fuchsian groups", by Svetlana Katok and am trying to solve one of the given exercises in chapter 2. The goal is to prove that every hyperbolic and parabolic cyclic ...
0
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0answers
12 views

Triangulating 2 connected points inside polygon

I'm trying to triangulate 2 connected points inside a polygon (triangles and quadrilaterals). Searching through the literature kept pushing me towards Delaunay triangulations with edge-flipping ...
4
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2answers
79 views

Why am I getting two different answer for this geometry question?

There is a right triangle $\textrm{ABC}$ like the diagram above, and the point $\textrm{D}$ set so that $\mathrm{\overline{AD}=\overline{BC}}$. If point $\mathrm{E}$ divides line segment $\mathrm{AB}$ ...
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0answers
8 views

Internal Coordinate System / Reference Frame to Align Set of Vectors?

There are sets of unit vectors $X_i = [x_{i,1}, x_{i,2}, ..., x_{i,N}], x_{i,j} \in \mathbb R^3$, where the order of vectors remains fixed, but the sets might be 3D rotations of each other Is there ...
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0answers
26 views

Is it true that re-parametrization of a curve does not change its length?

I'm trying to understand what is happening with the arc length parametrization, but it seems that something is missing. I considered the curve $a(t)=\left(e^{t}\cdot \sin(t), e^{t}\cdot \cos(t)\...
1
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1answer
33 views

Identification of two hypercubes in $\mathbb{R}^d$ [on hold]

Use the notation $k = (k_1,\ldots, k_d) \in \mathbb{R}^d$. How to see that the regions $$ \Big \{ \, \, k \in \mathbb{R}^d \, \, : \sum_{i=1}^d \cos(k_i) > 0 \, \, \Big \} \cap \Big \{ k \in \...
2
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1answer
71 views

Pair of straight lines parallel vs coincident

The equation $ax^2+2hxy+by^2+2gx+2fy+c=0$ represents a pair of parallel lines if $\dfrac{a}{h}=\dfrac{h}{b}=\dfrac{g}{f}$ and the distance between the parallel lines is $2\sqrt{\dfrac{g^2-ac}{a(a+b)...
1
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2answers
30 views

Volume of a frustum given the bottom radius and the top cone height.

A cone with base radius 12 cm is sliced parallel to its base, as shown, to remove a smaller cone of height 15 cm. If the height of the smaller cone is three-fourths that of the original cone, what ...
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0answers
24 views

Orthogonal rotations in higher dimensions? [on hold]

Rotations in higher dimensions are given by a parametrized $n\times n$ matrix. I want to parametrize so that the rotations are orthogonal. How can this be done? I want explicit parametrization.
1
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1answer
60 views

Long Geometry problem

Let $ABC$ be a scalene triangle and let its incircle touch sides $BC$, $CA$ and $AB$ at points $D,E$ and $F$ respectively. Let line $AD$ intersect this incircle at point $X$. Point $M$ is chosen ...
0
votes
1answer
21 views

Mathematical notation for orthogonal projection of a set of points on a line

How could i mathematically denote the following: Assume a matrix P which represents the coordinates of a set of points (each row = a single point). Each point in the matrix P is projected on a line ...
8
votes
4answers
215 views

A challenging geometry proof?

Given: $C$ on $\overline{AB}$ such that $BC=3AC$ and $m\angle B=2m\angle XCB$. To show: $AX=2AC+BX$ I have verified this result with trigonometry and analytic geometry and double-checked my work with ...