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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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Lost in a Forest Problem 3d Verision: Point and Plane

The lost in a forest problem is famous, specifically see this problem. The problem takes place in the plane. Consider the following 3d version. There is a point $p \in \mathbb{R}^3$ and a plane $P$ ...
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1answer
35 views

An Interesting Curve with Geometry and Calculus [on hold]

Given an ellipse (in blue) with a horizontal major axis of length 10 and a vertical minor axis of length 6, we construct a curve (in red) which is the boundary of the convex hull of points which are 3 ...
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1answer
12 views

Regularity of the surface

There is given curve parameterized by arc length $ \alpha (s), s \in I $, and there is the surface $ r(s,v)=\alpha (s) + vT(s) , s\in I, v \in \mathbb{R}$, where $T(s)$ is the tangent vector field. I ...
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1answer
39 views

Why can not we know if the rhombus are similar?

I have this statement: Are two similar rhombuses, if they have the three corresponding pairs of angles equal? My answer was yes. Following the fundamental theorem (AA), in case of the triangle ...
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0answers
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Gaussian curvature under a conformal map

This is an exercise from GTM 275, Differential Geometry by Loring W. Tu, page 94. Let $T:M \rightarrow M'$ be a diffeomorphism of Riemannian manifolds of dimension 2. Suppose at each point $p \in M$, ...
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1answer
45 views

Bounds of rotated egg?

I'm attempting to find the bounding box for a rotated egg shape without resorting to interpolation if at all possible. In researching this I've come across this link which lists useful equations for ...
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1answer
26 views

Can a smooth closed curve's interior be split into smaller copies of itself?

Is there a smooth Jordan curve whose interior can be split into any finite number of regions whose boundaries are similar to the original curve? E.g, a square can be split into four smaller squares, ...
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2answers
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center of gravity given four weights

I have this problem. I have four points that corresponds to four vertices of rectangle. Every vertex has a load cell that return a weight. How can I calculate the center of gravity, in a range of 0-...
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How can you find the degrees of a curved line's corners by graphing a vertical and horizontal line?

Sorry if my title appears subjective I am sure there is a more technical term for what I am asking. Maybe the image will explain my question better.
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2answers
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How to determine if the curve is closed

There is a given curve $ α(t) = (3 \cos t − \cos 3t, 3 \sin t − \sin 3t)$ and I have to determine if it's a closed one. I tried to find the answer and failed. What should I do? Thanks.
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1answer
27 views

Calculating volume of a bell shaped container

Given $r_{b}$, $h_{b}$ and $h_{t}$, what would be the equation for calculating the volume of used space in a bell shaped container. Example sketch Example sketch What I have so far: The shape can ...
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0answers
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Converting $X, Y$ on plane to world $X, Y, Z$ coordinates

I working on program for displaying 3-dimensional objects on the computer screen using Ray Tracing method. And now I came across a problem with throwing rays from the orthogonal camera. So, as you ...
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3answers
31 views

Angle of sector formed by cutting a cone

A cone has a height of 10cm and circular base with radius 4.it is slit and cut open to form a sector.find the angle formed by the two radii.which is the simplest method to solve this?
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2answers
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How to find equation of a plane that is passing through 2 points and normal to other plane??

The given points are $M(3,-1,2)$ and $M1(0,1,2)$ , plane A is passing though this 2 points. Plane B $2x-y+2z-1=0$ and its normal to A. How to we find the equation of plane A? I have tried with cross ...
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1answer
36 views

A question concerning $R(n) :\equiv (\exists x,y,z)\ x^2 + y^2 = z^2 \wedge x + y + z =n$

A square number is a number $n$ with the property $S(n) :\equiv (\exists x)\ x^2 = n$. A number $n \neq 0$ is a square number iff it has an odd number of divisors. A hypotenuse number is a number $n$ ...
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Euler Characteristic of Spherical Polygon

I am a little confused on what the Euler Characteristic of an n-sided spherical polygon would be. Because the polygon is 3 dimensional, would it be considered to have 2 faces or 1? Following that, I ...
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1answer
40 views

Geometric interpretation of $\{z\in \mathbb{C} : \text{Im}(\frac{z-a}{b})>0\}$

I'm trying to find a geometric interpretation for the set mentioned in the question where $a$ and $b$ are given complex numbers that are not zero. What I've done so far: $$\text{Im}\frac{z-a}{b}=\...
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3answers
67 views

Geometric construction of a triangle, provided an angle, an internal angular bisector of this angle, and the length of the side opposite to this angle

How can I construct a triangle knowing the following information below? A) An angle. B) the length of the internal bisector for the given angle. C) the length of the side opposite to the given ...
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0answers
22 views

Understanding whether or not a curve is a characteristic projection

I am trying to familiarise myself with the method of characteristics in solving first-order PDEs. I have come across the following example: Solve the first-order PDE $$\frac{\partial u}{\partial x}+\...
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3answers
71 views

Equation for great circles of a sphere [on hold]

Background: I have a sphere of some radius R. What I'm trying to do is essentially create a wireframe consisting of great circles that run along the sphere. I want an equation to represent the great ...
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3answers
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Is a $3D$ volume possible with only $3$ faces?

I was wondering if a $3D$ volume with only $3$ faces was possible. I know that in the Euclidean space, it is technically not possible (the minimum being $4$ faces), but maybe there was some other way. ...
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How many whole pieces can be taken out in this way? (Infinite chocolate bar problem)

The animation above implies that we can eat an infinite amount of chocolate from the same chocolate bar, but it is misleading—after each reassembly of the chocolate bar, the height of the chocolate ...
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2answers
42 views

How do I figure out the proportionality between two aspect ratios

How do I find the dimensions that correspond to an aspect ratio, in the same way that 800x1280, for example, corresponds to 5:8??? seemed like a simple enough problem, until I sat down and tried to ...
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1answer
29 views

How are these cross-product summations equivalent?

Trying to determine how the $X_{i+1}$ is no longer applicable by changing summation bounds: $$\sum_{i=0}^{n-1} (X_{i} + X_{i+1})(Y_{i+1} - Y_{i}) = \sum_{i=1}^{n} X_{i}(Y_{i+1} - Y_{i - 1})$$ Can ...
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1answer
181 views

Maximum number of concave vertices in a polygon

Let, $P$ be a closed polygon with $10$ sides and $10$ vertices. Let,$k$ be the number of interior angles of $P$ greater than $180^\circ$. Then the maximum value of $k$ is- All I could make out is ...
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1answer
48 views

What is the term for a line that doesn't touch a function?

A line that cuts into a function is a secant line, and a line that just touches a function is a tangent line. But what is the term for a line that does not touch the function? Take the parabola: $$y=...
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1answer
54 views

Finding an angle in a triangle formed by constructions with angle bisectors

AD, BF and CE are the angle bisectors of the $\triangle{ABC}$. $\angle{BAC}=120°$. How to find $\angle{EDF}$?
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1answer
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Length of side of maximum volume hypercube inside hypersphere

Suppose we have an $n$ dimensional hypersphere of radius $R$. What will the length of a side of the hypercube be, that has maximum volume?
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2answers
94 views

A little conjecture about a circle related to any triangle

Given any triangle $\triangle ABC$, let denote with $D$, $E$ and $F$ the midpoints of the three sides, and draw the three circles with centers in $D,E,F$ and passing by $A,B,C$, respectively. These ...
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What are the symmetries of a Trigonal trapezohedron?

The asymmetric version of a Trigonal Trapezohedron is supposed to be a fair die just like a cube, meaning I can start with one face and rotate it about the center of the solid to get each of the other ...
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2answers
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How to inscribe a quadrilateral in a circle?

In the case of the isosceles trapezoid, I have seen drawings in which its longest base is always the diameter of the circumference. But, how can I know that the longest base will match the diameter? ...
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Need help with this SEA Math Competition question! Geometry

Two of the altitudes of a scalene triangle with integer-length sides are 6 and 14 units. Given that the third altitude is also an integer, what is the maximum it can be? This was a multiple choice ...
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1answer
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Tangential quadrilaterals description

Take a quadrilateral $ABCD$ and consider a line parallel to each of its side such that: 1- the distance between the parallel line and the side is some fixed amount $x$ and 2- the parallel line are "...
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Decompose a parametric curve into monotone segments

I'm looking for a numerical computational method to decompose a parametric curve $C(t)$ by subdivision into xy-monotone segments. I searched the web thoroughly, but didn't find any usable relevant ...
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1answer
37 views

Derivation of formula to calculate the angle of inclination

I understand that in order to find the angle between two lines, you can calculate the angle of inclination of each of the lines and take the difference between those two. However, I'm not sure exactly ...
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5answers
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straight line, the question is- Find where the line joining the points $(-3,5)$ and $(-4,8)$ meets the line $x=15$.

the question is- Find where the line joining the points $(-3,5)$ and $(-4,8)$ meets the line $x=15$ . all i could do was work out the gradient $-3$ please help!! im not sure how to work this or if ...
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Formal term for sets of line segments [on hold]

What is the name for all sets of line segments defined by n points equidistant from a common center? For example, three such points may define three line segments (an equilateral triangle), four ...
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2answers
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find the coordinates of the area of the largest isosceles triangle that can be inscribed in an ellipsoid

for the given ellipsoid $$x^2+ \frac{y^2}{4}=1$$ I have to find the coordinates of isosceles triangle that can be inscribed in this ellipsoid. the basis of the triangle is parallel to x axis My ...
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1answer
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Hard trigonometry problem using basic theory [on hold]

The angles $A$, $B$, $C$ , $D$ of a convex quadrilateral satisfy the relation: $$\cos A+\cos B+\cos C+\cos D = 0$$ Prove that $ABCD$ is a trapezium or is cyclic. Please note: I have been told to ...
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0answers
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Is it always possible to cut disjoint thirds of two rectangles?

There are two rectangles, red and blue. Is it always possible to cut two disjoint rectangles, one containing 1/3 the area of red and one containing 1/3 the area of blue? The following figure shows ...
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2answers
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Not able to understand the solution of a simple math problem

I was practicing the following beginner level problem at codechef Santosh has a farm at Byteland. He has a very big family to look after. His life takes a sudden turn and he runs into a financial ...
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0answers
62 views

Plane geometry as an area of research; advanced courses available? [on hold]

A recent question involved the geometry of a cyclic quadrilateral. I had to google it to find out its properties (Wikipedia). During my high school education I had plane geometry (tenth grade) where ...
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1answer
37 views

I am not understanding this math about alternate angles

This is the math. This is the given solution :- This and this Now, my problem here is, $\angle ABC$ and $\angle DAC$ are supposed to be alternate angles here, and thus equal. But as far as I know,...
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1answer
29 views

Proving that the cone is isometric to the plane

There is a given cone, $ 3z = 4 \sqrt{x^2+y^2} $. I have to prove that this cone is isometric to the plane. First, I parametrized the cone: $ r(u,v)=(u \cos v, u \sin v, \frac{4}{3} u) $. The ...
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2answers
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Is “alignment” (or something else) a better word than “direction” for a senseless direction?

When I encountered the concept of direction being prior to the ordering of points on a line representing (parallel to) the direction, I thought it was a valuable distinction. I resolved to keep this ...
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1answer
26 views

Find the angle between parallel lines

I am trying to solve the following problem. Given: $$\overleftrightarrow{AB} \parallel \overleftrightarrow{DE}$$ And the measures of angles $$\angle BAC = 42 ^{\circ}$$ $$\angle EDC= 54 ^{\circ}$$ ...
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2answers
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Intuition for convex hull

For some set $V \subset [a,b]^d$, define the convex hull of $V$ as the set $$\{\lambda_1v_1 + ... + \lambda_kv_k: \ \lambda_i \ge 0, \ v_i \in V, \ \sum_{i=1}^k \lambda_i = 1, k = 1, 2, 3, ...\}.$$ ...
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1answer
83 views

Proving radius of circle $\dfrac{\triangle}{a}\tan^2\dfrac{A}{2}$

If a circle be drawn touching the inscribed and circumscribed circles of a $\triangle ABC$ and the side $BC$ externally, prove that its radius is: $$r=\dfrac{\triangle}{a}\tan^2\dfrac{A}{2}$$ I tried ...
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26 views

Ratio of triangle to that formed by angular bisector.

If the bisectors of the angles of triangle $ABC$ meet the opposite sides in $A',B', C'$ prove that ratios of the area of the triangles $A'B'C'$ and $ABC$ is $$2\sin\dfrac{A}{2}\sin\dfrac{B}{2}\sin\...
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30 views

Geometry Circles [on hold]

If C and C' are two circles in the plane and C is tangent to C' on the inside of C', which of the following statements is necessarily true? (A) Whether or not C passes through the centre of C' ...