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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What is the answer [closed]

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Given Circle $E$ inscribed in square $ABCD,$ prove that its radius is equal to half the length of the square's side length.

Given Circle E inscribed in square ABCD, prove that its radius is equal to half the length of the square's side length. I'm stuck on this proof but I've got a few ideas. If anyone could help that ...
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Applications of Vector Geometry

Current in maths, we are learning vector geometry. Specifically, we are trying to express lengths on a diagram ( eg. CE ) as vectors in terms of vectors that are already given. For example: Example ...
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How Can I Calculate Irregular Polygon's Internal Angle? [closed]

For example there is irregular polygon and I choose a vertex and I want to calculate it's interior angle, the angle can be more than, less than or equal 180 degrees. How can I calculate and how can I ...
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In $\triangle ABC$, there are segments $MN||AC$ and$NP||AB(M∈AB; N∈BC; K∈AC). AM:MB=7:2.$ Find $KC:AK$.

I have come to the conclusion that KC:AK=7:2, but I don't know how to prove it.
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1 vote
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Why would one have to assume that the rate of evaporation has to be proportional to the surface area?

The problem states the following: A container in the shape of a right circular cone with vertex angle a right angle is partially filled with water. A. Suppose water is added at a rate of 3 cu.cm/sec. ...
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Prove Similarity Ratio? [closed]

If two triangles ∆ABC and ∆PQR are similar, then Prove That : AB/PQ = BC/QR = AC/PR Thats it Any help would be admired.
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1 answer
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Convert a direction into an angle on our compass

I have a new position vector and old x and z coordinates and I need to determine the angle traveled. So far I am getting the slopes absolute value and then using that to generate an angle. Based on ...
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Derive the Pythagorean Theorem from the Taylor series for trig functions

Inspired by this recent question. Starting with the function definitions: $$C(x) := \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$$ $$S(x) := \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$$ ...
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-6 votes
1 answer
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1. Prove that the sum of the angles of any triangle PQR is two right angles. [closed]

A. Prove that the sum of the angles of any triangle PQR is two Right angles. B. In triangle PQR ,X is a point on line QR such that angle P =angle Q. Prove that angle X=angle P
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Help with Geometry Problem, find the radius of cylinder in the figure?

This problem was given to me by a friend of mine who is a senior in high school in Mexico. The goal is to find the radius of the bottom-leftmost circle that is inscribed within the cylinder. The ...
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Realizing the Hopf fibration as the phase space of a small oscillations spherical pendulum

Page 35's of Arnold's ODE book "The deviation from the vertical is characterized by two numbers $x$ and $y$. It is known from mechanics that the equations of small oscillations have the form: $\...
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2 votes
1 answer
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Algorithm to find the angle of a direction

I need to translate the distance between two points into an angle from 0 to 359. To do this I use the new position coordinates which is defined by a vector and subtract the original position. This ...
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Solid angles of parallelepiped. [closed]

Can we in easy way proof that sum of solid angles of parallelepiped equals 4$\pi$?
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2 answers
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How to find the points that are in-between 4 planes

I have a set of points, I want to select only a few of those points. For that, I have 4 planes equations in the general form and I want to be able to check in a look if a given point would exist in ...
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For the reflection $\phi$ in the sphere $S(a,r)$ it holds that $\phi(B^n) = B^n$ if and only if $\phi(a*) = 0$

This is a question about Theorem 3.4.2 in "Geometry of discrete groups" from Beardon. Let $\phi$ be the reflection in a sphere $S(a,r)$ for $a \in \hat{\mathbb{R}^n}$, $r \in \mathbb{R}$. ...
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1 answer
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Sum of "angles" of a 3D tetrahedron

We know that the sum of angles of a triangle equals the straight angle (180 degrees). Can we convert a 2D theorem to 3D? e. g. We can generalize the triangle to a tetrahedron, angles of the triangle ...
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1 answer
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Find the area ratio of the triangular regions $ATK$ and $LKS$ .

For reference: If, $T$ and $K$ are points of tangency, measure of the $\overset{\LARGE{\frown}}{AB}$ is twice the measure of the $\angle ASL$, $BS=3$, $KS=1$ and $\frac{TB}{4} =\frac{AT}{3}$ ....
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Projective Basis Explanation, Visualisation

I have trouble understanding the geometric idea behind a projective basis. So for example in $\mathbb{P}^2(K)$ we have $[1:0:0],[0:1:0],[0:0:1], [1:1:1]$ but why? I know that $$ [1:0:0] = k\cdot \vec{...
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2 votes
0 answers
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A little trouble in proving that geodesic flow on Riemannian manifolds with bounded negative sectional curvature is Anosov (from Klingenberg's book)

I'm dealing with the proof from Klingenberg's "Riemannian Geometry" of the Anosov theorem on geodesic flows on complete Riemannian manifold with bounded negative sectional curvature, pages ...
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Given two points, construct a parallelepiped with a square base(length given)

So I have been doing this for a while now, and I am afraid I am horribly overthinking this problem, so I came here for some fresh takes on this. I want to obtain all the vertices of a parallelepiped ...
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-2 votes
1 answer
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Distance in arc length along the circle [closed]

In my notebook appears $d$ = distance in arc length along the circle. What does it mean??
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1 vote
1 answer
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Limits of abstract smooth surfaces

For $r >0$, let $K_r \subseteq \mathbb{C}$ be the closed subset, $K_r = \mathbb{C} \setminus D(0,r)$. Define $S_r$ to be the quotient of $K_r$ under the identification: $$ z \sim -z, \hspace{1cm} ...
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Grasshopper problem regarding jumps across the real number line

"2021 grasshoppers are placed on the real line (not necessarily in integer points). At every step, one of the grashoppers hops over another one, landing on the opposite side at the same distance. ...
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Finding distance from a point to a set

What is $d_A(p)$, where $A := \{(x,y) \in R^2 : x^2 + y^2 = 1\} ?$ Find an explicit expression. Also $p=(a, b) \in R^2$ This is a question from the book: Topology of metric spaces by S. Kumersan. I ...
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4 votes
1 answer
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Does this locus have a name?

This comes from variational analysis and in particular from the definition of the tangent cone of a set. Say we have a circle centered at $(0, 1/\tau)$ with radius $1/\tau$, $\tau > 0$. All of ...
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How can I create an Arc path from 3 point (p1-->p2-->p3) in 3D space?

I want to create an Arc path over 3 points in 3D space. I decided to draw a circle over them and then extract sector from p1 to p3 that passes through P2. I use this code but it did not extract sector ...
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1 vote
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Rotating about non-centered axis in 4-d

In 3 dimensions, there is a concept of rotating about an arbitrary line that is not centered at the origin. To pull this off, we first move the origin so that it is on this line. Anywhere on the line ...
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6 votes
3 answers
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Analytic definition implies geometric definition of trigonometric functions

It is well known how we can arrive to the power series definition of trigonometric functions starting from their definition in terms of the unit circle. I'm trying to do the converse, i.e. start from ...
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Counterpart of axis-angle rotation matrix in 4 dimensions?

In 3-dimensional space, we have an explicit formula for the rotation matrix which will rotate about a vector $\vec{a} = [a_x, a_y, a_z]$. This is given by: $$ \begin{bmatrix} \cos\theta+a_x^2(1-\cos\...
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1 vote
1 answer
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Find the area of ​the shaded region.

Given, $MT=2$ and $AC=8$. Calculate the area of ​​the shaded region. $\triangle BMT_(notable)\implies (a, 2a, a\sqrt5)\\ \therefore MT = 2, BT=4\\ \triangle ABF \sim \triangle MBT \implies k = \frac{...
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3 votes
1 answer
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How many circles pass through 2 points but also tangent to a given circle?

Given: a circle $O$. and $2$ points $A, B$ out of that circle. How many new circles that are tangent to circle $O$ can we form which also pass through points $A,B$. My Geometry is a somewhat rusty - I ...
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3 votes
1 answer
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Area of a special triangle in a quadrilateral

As is the case with triangles, I am sure that there is an entire ocean of not-so-well-known theorems about Euclidean quadrilaterals. In particular, I am interested in the following problem and suspect ...
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2 votes
0 answers
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Finding an envelope for a moving circular sector

Preamble: I want to find the curve which bounds a moving circular sector, i.e. an envelope for the following family of plane curves. Suppose that we are given a "perspective" point $T$ and ...
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1 vote
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Can we determine an $3D$ object uniquely if we know its projections from all views?

I know that it's not enough if we use its plan, front and side views only. But what if we know its projections from arbitrary views? Non-visible sharp edges and silhouette boundary will be drawed as ...
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4 votes
1 answer
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Can mathematics distinguish left and right?

Imagine, a mathematician from another galaxy lands on the earth. Is there a way we can explain to him what is "counterclockwise" without showing him a picture? Things like Green's formula, ...
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1 vote
2 answers
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Cannot find the x and y angles. Is there something about the congruent lines in the two triangles that i need to use to find the answer?

Got the triangle on the left as 60° 60° 60° and the middle angle at the bottom is 46° but that's about it. What am I missing to get X and Y?
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1 vote
0 answers
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Showing Euclid's Proposition 30 ("Lines parallel to the same line are parallel to each other") is equivalent to the 5th Postulate.

I am trying to show that the 30th Euclid's proposition, "Straight lines parallel to the same straight line are also parallel to one another." is equivalent to the 5th Postulate: "If ...
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2 votes
1 answer
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Finding angle in circle to produce equal areas

I have a circle that is divided into 4 quadrants with a vertical and a horizontal axis. The center of the circle (where the axes cross) is point b. The top of the vertical axis is point d. On the ...
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0 votes
0 answers
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Unit speed torsion and curvature question

Let $γ$ be a unit speed curve in $R^3$ with curvature $k > 0$ everywhere. Let $τ$ be the torsion of $γ$ and assume that $\fracτk$ is constant. Prove that there exists a unit vector $a ∈ R^3$ such ...
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0 votes
1 answer
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Mapping an arbitrary 4 node quadrilateral element in space (x,y,z) to unit square

I need to integrate a function G(x,y,z) over an arbitrary 4 node quadrilateral element (4 nodes xi,yi,zi on a same plane). I know how to handle it when the 4 nodes (xi,yi) are on the X-Y plane, i.e., ...
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-5 votes
0 answers
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Proving two concurrencies and a collinearity from a configuration of circles [closed]

A hard but fantastic geometry problem: $\Omega$ (with center $O$) intersects with circles $\omega _{1}$, $\omega_2$, $\omega_3$. $\odot O_{a1}$ and $\odot O_{a2}$ are circumscribed with $\omega _2$, $...
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0 votes
0 answers
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Parametrically enlarge one ellipsoid to fit another one

I'm trying to figure out the smallest enlargement factor which I need to apply to one ellipsoid $E_1$ in order to fit another one $E_2$. Precisely, let $E(c, S) := \{x | (x-c)^T S (x-c) \leq 1\}$ be ...
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-9 votes
0 answers
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Solve the problems if you can [closed]

please help me solve this problem and give me a correct answer, only if you can
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0 votes
1 answer
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Largest area with given perimeter, one straight edge

A common example to introduce quadratic functions is to ask for a rectangle with the largest area when the perimeter is given and you are allowed to use one additional edge that does not count towards ...
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1 vote
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Basic Geometry, extract relation between sides and angles

I am asked to extract the geometric relation from this figure: $ABC$ is not a right triangle. angle $A=\theta_1+\theta_2$. The only thing I thought about is the cosine rule: $$BC^2=x^2+z^2-2xz\cos(\...
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Usage of similar triangles

This is typically a general question about when we could use similar triangles in real life. Googling this question let me understand that it is very possible to use them for trees, buildings heights ....
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4 votes
0 answers
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Exercise related to vector fields, map degrees and Poincare-Hopf's Theorem

I got stuck with one exercise from Chapter 3.5 in Guillemin and Pollack's book, which I used to study differential topology by myself: Given a vector field $\overrightarrow{v}$ with isolated zeros in $...
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1 vote
0 answers
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Embedded arcs staying embedded after squaring.

Let $a : [0,1] \to D^2 = \{z \in \mathbb{C} : |z| \leq 1 \}$ be a continuous map with $|a(0)| = |a(1)|$ and further assume that $a$ is an embedded - so $a$ is an embedded arc. Let $f : D \to D$ given ...
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0 votes
1 answer
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Fitting multiple plane segments through a set of points

Imagine a 2D space in which are samples. Each sample belongs to a class and there are two classes "true" and "false". In this problem, we know that these two classes are separable ...
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