# Questions tagged [geometry]

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

31,926 questions
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### 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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### Gluing $2n$-gons

I am confused about this how I would visualize/prove this. Given that we have a regular $2n$-gon that preserves orientation (the surface is orientable because the $2n$-gon's opposite sides are glued ...
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### 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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### 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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### 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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### 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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### 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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### How to prove that the statement “there exists a constant, X, such that the sum of all angles in a triangle is equal to X” is equal to postulate 5 [closed]

I have this as a homework question, and didn't know if it would be the same proof as when the angles of a triangle sum to 180. I was also unsure if I should be using theorems like exterior angle that ...
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### 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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### 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 $AB$...
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### Is there any proof that corresponding angles are equal? [closed]

Can you prove that corresponding angles are equal?
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### Generating function of cuts of some graph

Consider the graph $G$, each edge of which has weight $T$. Also consider some cut of $G$. We call the weight of the cut - product of all edges included in the cut. Generating function of cuts of $G$ ...
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### Is it true that if inclusion of a boundary component is a homotopy equivalence then the manifold deformation retracts onto it?

In topological spaces, if $A\subset X$ and the inclusion is a homotopy equivalence, that doesn't imply that $X$ deformation retracts onto $A$. For example take one of those examples of a contractible ...
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### What is the induced orientation on a 1-manifold with boundary that is the 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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### Chess : how to count all legal moves on diagonals for a queen placed in $(x,y)$

I'm trying to get a formula that counts all legal spaces over diagonals for a given point $(x,y)$, e.g.: In the above image, you can see there are $13$ legal squares over diagonals. Is there a ...
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### Is there a better name for a “spherical annulus”?

For example, the set $\{x\in\mathbb{R}^d: 1\leq|x|\leq2\}$. For $d=2$, I'd call this an annulus. What do we call it in higher dimensions?
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### An equilateral triangle has a perimeter of 810 cm. Calculate the area of ​an octagon that has the side equal to 4/9 of the side of the triangle. [closed]

An equilateral triangle has a perimeter of 810 cm. Calculate the area of ​​an octagon that has the side equal to 4/9 of the side of the triangle.
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### Proposition 3.14: Geometry

Please help prove the following proposition: Proposition 3.14 Supplementary angles of congruent angles are congruent. Is this right? (1) Suppose angle ABC is congruent to angle DEF (given) (2) We ...
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### Introduction to Geometry Books

I am looking for a book that covers introduction to geometry. Currently, I am reading "Geometry: A metric approach with models", by R.S Millman. I like the book but I would like to read another highly ...
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### Exact Differential Equation Geometry

In a variety of contexts, I have noticed hints of a strong connection between exact differential equations and machinery from multivariable calculus. From another question, I have gathered that the ...
Let $f$ be an isometry of the Poincaré half-plane model of two-dimensional hyperbolic geometry, denoted by $\mathbb{H}^2$. Prove that if the distance $d(z,f(z))=c$ for some constant $c\geq0$ for all \$...