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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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Dihedral angles of a simplex

Given a $k$-simplex $(p_0, ..., p_k)$, where $p_i$ are $n$-dimensional points. Define the dihedral angle $\theta_j$ as the angle between the $(k-1)$-facets incident to the $(k-2)$-facet $e_j$. Now, ...
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Let $m$ be the length of the diagonal of a regular four-sided prism which closes an angle $\alpha$ with the side of the prism

Let $m$ be the length of the diagonal of a regular four-sided prism which closes an angle $\alpha$ with the side of the prism. Calculate the surface of the perimeter. I guessed the angle is the ...
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Make a formal description of a volume

I'm attempting to write formally the description of a volume. For an academical definition for engineers. Can you give me your feedback if it is correctly defined? In simple terms, it is a cylinder ...
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1answer
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Length of tangents of incircle and excircle

Given a triangle $ABC$ , the incircle touches side $BC $ at $D $ and the excircle touches the side $BC$ at $F$ . Prove that $BF=CD$ . Can't think of a way to relate the tangents of the incircle and ...
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Challenging Problems in Geometry 4-24 challenge

So I have been working on this geometry problem in the book mentioned in the title, and it doesn't have solutions to all the problems. I'm stuck, any help is appreciated. Two circles are tangent ...
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Find the equation of circle with given equation of chords and its length

Question: Line $l_1:2x-3y+2=0$ and $l_2:3x-2y+3=0$ is the chord of circle with length $26$ and $24$ units respectively. Find the equation of circle. Is the solution unique? If not unique, what is the ...
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Parameterize an implicit equation

I am trying to parameterize the following implicit equation so that I can sketch it. $x^2+y^2=1+4.5\sin^2(xy)$ However, I am having problems with parameterizing the equation because of the $xy$ and ...
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1answer
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How do I find the radius of one circle given constraints based on another intersecting circle?

I have two circles which intersect in such a way that the overhang of the smaller circle is a percentage of that smaller circle's radius. How do I define the function of the radius of the smaller ...
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1answer
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Vertical cut-off distance to transform vertical parabola into rotated circle segment

Given a known vertical, negative parabola ($y=ax^2+bx+c$ with $a$,$b$ & $c$ known parameters): how far down from the vertex of the parabola (the top) is the vertical distance, so when you "cut if ...
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1answer
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An interesting (conjectural) property of any triangle

Given any triangle $\triangle ABC$, we can always build three ellipses, each of them having foci in two of the vertices and passing through the third one, as shown in the following picture: In ...
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2answers
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Rotation transformation between two frames

My goal is to express the transformation between the black frame F1 and the other one F0 (Green, Red, Violet): All what I know ...
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1answer
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Prove or disprove: Every polygon with an even number of vertices may be partitioned by diagonals into quadrilaterals

Question: True or False? Every polygon with an even number of vertices may be partitioned by diagonals into quadrilaterals. Details: Any orthogonal polygon may be partitioned by ...
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Is the definition of a Point in Geometry consistent? [on hold]

A point is defined to have length = 0. Surely such a definition is self contradictory; anything with length 0 cannot exist. The existing definition yields nonsense; for example, how many points on a ...
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1answer
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Vectors and 3d Geometry [on hold]

If a,b and c are the vertices of equilateral triangle and the orthocentre is at the origin then, which of the following is True? $a+b+c=0$ $a^2+b^2=c^2$ $a+b=c$ None
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Convex quadrilateral from many points

Does anyone know how to get (enclosing)‘convex quadrilateral’ from randomly scattered points? The solution must have minimal area among the all possible quadrilaterals. Any suggestion or conjectures ...
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How do I “normalize” two sample of values while keeping their ratio?

The values are floating point numbers and can be negative. What I would like to do is make them fit into an interval ( in my case it's [ 0 ; 400 ] ). To give more details about my problem, I have two ...
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Prove that $SD'$, $EF$and $HI$ are concurrent?

Let $\triangle ABC$ be a triangle with incenter $I$ and orthocenter $H$. The incircle of $\triangle ABC$ touches $BC$, $CA$, $AB$ at $D, E, F$ respectively. Let $D'$ be the reflection of $D$ through $...
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How to find the closest intersection between a line and several spheres without using a square root?

I am doing some ray tracing and I have the equation of a line and several spheres with their coordinates and radius. I want to find the closest intersection with a sphere, which I know I can do with ...
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Proving that $x^2+2y^2+3z^3 = 1$ is an embedded manifold

I am working on the following exercise: Consider $S = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + 2y^2 + 3z^3 = 1 \text{ and } z>0\}$ Show that $S$ can be parametrised as a graph of a function from an ...
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1answer
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How to perform a linear fit of a path in the plane?

I have an ordered set of $n$ points $(x_1,y_1),\dots,(x_n,y_n)$ and I would like to choose an ordered subset of these points, say $(x_{k_1},y_{k_1}),\dots,(x_{k_s},y_{k_s})$, with $(x_{k_1},y_{k_1})=(...
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2answers
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$AH=AS$ where $H$ is the orthocenter of $\triangle ABC$ and $S$ is the midpoint of the arc $BHC$ of the circumcircle of $\triangle BHC$

The altitudes of an acute triangle $ABC$ which is not isosceles concur at the point $H$. Let $S$ be the midpoint of the circular arc $BHC$ of the circumcenter of the triangle $BCH$. If $AS$ and $AH$ ...
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1answer
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How did I obtain incorrect results for $P_{B\leftarrow C}$ and $P_{C\leftarrow B}$ geometrically?

I'm given two bases in $\mathbb{R}^2$: $B = \{\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} -1 \\ 2 \end{pmatrix}\}$ and $C = \{\begin{pmatrix} -4 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 5 \...
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1answer
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Geometrical meaning of intersection between line and bundle of circles

Given the following bundle of circles: $$x^2(1 + k) + y^2(1 + k) + x(24 + 8k) + y(-4 -4k) + 4k + 132$$ I would have to find the circles tangent to the x-axis. The way I would solve this is: plug $...
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3answers
49 views

help to find radius

How do I solve this problem mathematically? What is the radius?? Knowing an arc length with a subtended chord (or opening) of known length. It will look like the letter "C". Say an arc length of 8 ...
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Can all points in a given plane be regarded as set elements of the plane?

When asked if a point P is contained in the plane α, is it correct notation to write that P∈α if my calculations conclude that the point is in the plane? My question is if this is the correct notation....
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1answer
20 views

Intersecting Angle between 2 Line Segments

I asked a similar question here before, which I'll link to here. I realized, however, that I needed to extend the question from the angle between a point and a line segment to the angle between 2 line ...
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1answer
36 views

Finding Bevel Angle

I need to find the bevel angle of a "box" with 4, 5, 6 or 8 sides. The shape is tilting outward at 10 and 20 degrees. All sides are the same length, and the bevel angle should be the same for each. ...
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3answers
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In a rhombus $ABCD$, prove that $IG\perp IP$. [on hold]

Let $ABCD$ be a rhombus with $\angle ADC=60^\circ$ (picture in attach file). The points $E$, $F$, $G$, and $H$ are midpoints on sides $AB$, $DA$, $CD$, and $BC$, respectively. Let $J$ be the ...
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1answer
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A conjecture involving three parabolas intrinsically bound to any triangle

Given any triangle $\triangle ABC$, we can build the parabola with directrix passing through the side $AB$ and focus in $C$. This curve intersects the other two sides in the points $D$ and $E$. ...
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1answer
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Can the computer be able to design a non-periodic tiling?

Non-periodic tiling is an interesting topics in geometry and beyond. PT (Penrose Tiling) is a famous example of non-periodic tiling. Rogers Penrose once said that the computer will stumble when ...
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Blindly removing inessential diagonals from a triangulation can lead to a bad convex partitioning

Assume that a simple polygon $P$ and a triangulation of it only using the diagonals is given. We say a diagonal $d$ is essential for vertex $v$ if removing $d$ creates a piece that is nonconvex at $v$....
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If ${\varphi}$ is angle between the tangent to the center of curvature of curve $c_1$ and principal normal of curve ${c}$ then [on hold]

$$tan{\varphi}=\frac{p}{(p^-){\sigma}}=\frac{(p)(T)}{p^-}$$ Where p denote the radius of carvature and T denote torsion,and $${\sigma}=\frac{1}{T}$$
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1answer
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An ellipse intrinsically bound to any triangle

Given any triangle $\triangle ABC$, we build the hyperbole with foci in $A$ and $B$ and passing through $C$. The hyperbole always intersects the side of the triangle that is opposite to the vertex ...
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isometries of angles

Find an isometry that maps ∠(1,1)(3,2)(2,2) to∠(3,−1)(3,2)(4,0) This is a problem for school but I don't just want the answer I want to know how to understand it. I graphed it on wolframalpha. I can ...
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1answer
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Coordinates of a point on a circle - review

Given a rotation $θ$ and a radius $r,$ how do I find the coordinate $(x,y)$? I saw an answer like this: From the picture, it seems that your circle has centre the origin, and radius $r.$ The ...
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2answers
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A conjecture about the intersections of three hyperboles related to any triangle

Given any triangle $\triangle ABC$, we build the hyperbole with foci in $A$ and $B$ and passing through $C$. Similarly, we can build other two hyperboles, one with foci in $A$ and $C$ and passing ...
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1answer
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Prove that the set of points $P$ such that $PO_1^2-r_1^2=k(PO_2^2-r_2^2)$ is a circle.

Let $\omega_1$ and $\omega_2$ be circles with respective centres $O_1$ and $O_2$ and respective radii $r_1$ and $r_2$. Let $k\in\mathbb{R}\setminus\{1\}$. Prove that the set of points $P$ such ...
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Finding coordinates of points using distance between points.

I got a question in my homework which I can't solve. Here is the question: (I am not a native speaker so please explain step by step and clearly.) Point $C$ internally divides the segment ...
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3answers
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How can I name a geometrical entity similar to a plane but with finite length and width?

I was considering just using the name rectangle for representing the set of points contained in a 3D plane for a given rectangular area. I would like to know whether there's a more appropriate name. ...
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Finding the region surrounded by light emitted from the vertices of a triangle through construction

P, Q and R represent the positions of three radio beacons. P is 450 km from Q, Q is 475 km from R, and R is 300 km P. Signals from P have a range of 300 km, Q has a range of 350 km and R has a range ...
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Set of hyperplanes to polyhedron topology

Problem By intersection of a set of planes (in three-dimensional space) I constructed an edge-vertex topology structure (see image). I was able to reconstruct the polygons in this structure by ...
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Fundamental Neighborhood in Ordinary Differential Equations

Consider the ODE \begin{align} \dot x = f(x) \tag{1} \end{align} Let $x_0$ be a hyperbolic fixed point of $(1)$. Let $V$ be a neighborhood of $x_0$ in $W^s(x_0)$, where $W^s(x_0)$ is the stable ...
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Is it true that the number is divisible by $p$?

Question: Let $a, b, c$ be positive integers and $p>3$ be a prime ($ a$ isn't divisible by $p$). Consider a quadratic polynomial $P(x) = ax^2+bx+c$, and assume that there exists $2 p-1$ ...
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2answers
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Calculate Points in Right Angle to Line

Given two points(blue) A(1,1) and B(3,3) I need to calculate the coordinates of two new points(orange) which are in an right angle to line AB and a certain distance from point A. So far, I've ...
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Drawing a locus

Consider two given circles of radii $r_1$ and $r_2$ with centres $C_1$ and $C_2$. A point $P$ is such that $\frac{r_1}{r_2} = \frac{PC_1}{PC_2}$. I wanted to know how the locus of $P$ would look like. ...
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Why are incircle and circumcircle of regular polygons concentric?

Do the incircle and the circumcircle of a regular polygon have to be concentrical? Why? Can't there be an irregular polygon with that property?
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Fermat's Point applies to isosceles triangles

Fermat's Point applies to equilateral triangles. Recently as I searched isosceles triangles on Wolfram Mathworld, I learnt that the same principle applies to similar isosceles triangles. Besides ...
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$\lambda$-Lemma or Inclination Lemma in Dynamical Systems

I appreciate it if someone provide some intuitions for the $\lambda$-Lemma (Inclination Lemma) in dynamical systems? I am trying to think about a pictorial example, but I am having a hard time.
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Parametric equation and distance formula

Help with this assignment! For the past three weeks now, I've been battling with mathematics assignment. I've successfully solved some but these ones posed a great challenge to me Eliminate the ...
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Geometry Problem about area of triangle

In triangle ABC , M is the mid point of BC and N is the mid point of AM. BN when extended Intersect AC at D. If area of the triangle ABC is 20 sq units , What is the area of triangle AND. See pic for ...