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

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Non-Circular Ellipse Nomenclature

A circle a special case of an ellipse. Is there a general word to describe an ellipse that is specifically not a circle?
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2answers
52 views

Help with applying thales theorem

Given: $MN||AB$ And $MO:AO = 3 : 5$ Find $CN:NA$ Here is drawing: I'm sure I have to apply the basic proportional theorem, however I don't see how. The answer is $3:2$
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1answer
40 views

Solid geometry /Volumes

I Need Some Help About The Geometry Problem, Question : " Lets suppose that Lines l and l' and l'' have cut each other in point A; The B And B' Are two random points From Line l , C and C' Are random ...
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1answer
41 views

One of the diagonals in a hexagon cuts of a triangle of area $\leq 1/6^{th}$ of the hexagon

Problem: Show that, in a convex hexagon, there exists a diagonal which cuts off a triangle of area not more than one-sixth of the hexagon. My attempt: Suppose we have a hexagon $ABCD$. There are two ...
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3answers
58 views

Finding centre of sphere inscribed in tetrahedron

Given the tetrahedron with vertices defined by vectors $a=(-4, -3, 1)$, $b=(8,3,1)$, $c= (2, 6, 1)$, $d=(4,3,3)$, find the centre of the sphere inscribed in the tetrahedron. My train of thought: ...
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1answer
34 views

the area that a part of an ellipse consumes in a square of a discrete grid

Think about a discrete grid of unit 1, which means the grid consists of infinite number of squares whose area is 1. You can assign a coordinate to each square and one of them will have the coordinate ...
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1answer
34 views

How do you prove that a quadrilateral is a square with vectors?

The vertices of a plane quadrilateral are labelled $A, B, A'$ and $B'$, in clockwise order. A point $O$ lies in the same plane and within the quadrilateral. The angles $AOB$ and $A'OB'$ are right ...
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1answer
47 views

Volume of tetrahedron defined by four vectors

Given four vectors $a$, $b$, $c$, $d$, which are vertices of a tetrahedron, can we find volume by considering it as $1/4$ of the volume of the parallelepiped whose volume is $[(a-b) \times (b-d)]\cdot ...
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2answers
59 views

How does the way we define cos or tan have anything to do with degrees of the angle?

So sine of angle $A$ is just a ratio. It is the ratio of the length of the opposite or perpendicular of angle $A$ and the hypotenuse. Cosine of angle $A$ is also just a ratio. It is the ratio of the ...
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0answers
53 views

Why is the area of a rectangle given as $l\times h$? [duplicate]

While this may seem like a simple question, I have found it very difficult to answer. My question is "Why is the area of a rectangle given as $l\times h$?" We define area as the "size of a two ...
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1answer
30 views

Importance of the ratio of the focal length/ aspect ratio, versus aspect ratio in ellipse?

We are investigating the self-assembly of prolate ellipsoids, equal short axes, with different aspect ratios, $\rho$, but constant volume. Besides the energy and entropy which are involved in this ...
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29 views

Why we name one side as the perpendicular of an angle but does not actually define it?

If I have a right angled triange: $\qquad \qquad \qquad \qquad$ I was wondering why we name the sides like this? The base of $A$ kind of make sense. But the perpendicular of $A$ what relation does it ...
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1answer
33 views

How do i calculate the area of shaded region?

I wouldn like to find the area of shaded region which it's circulated by a triangle as show in the below picture ? Note: I tried to draw other circle arround triangle ,but it's seems hard to me to ...
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2answers
32 views

Finding a length (x) inside a circle sector given another length (y) and the arc length (s) [closed]

I am stuck on a problem and can not seem to find a solution, maybe someone here can help me or at least tell me if it is possible to solve? Please look at the figure: The problem is: Find the ...
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1answer
23 views

Determine the concavity of an edge

Hie guys while trying to determine the concavity of an edge I came across a post saying ...
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2answers
33 views

Angular Difference between two rotation matrices on XZ plane

As the title says, I have two rotation matrices, $ R_1 $ and $ R_2 $. Both are rotation matrices that transform from the origin coordinate system $O$ to positions $1$ and $2$ (ignoring any ...
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0answers
87 views

Why does the Ellipsograph/Trammel of Archimedes draw an ellipse, really?

Here's a diagram of the device I mean, hard at work drawing an ellipse. I find this quite surprising, and would like to get to the bottom of things. Essentially, a rod (black line in animation) is ...
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0answers
43 views

There is a square that vertices are (0,0) (0,2) (2,0) (2,2) [duplicate]

A point P satisfies following condition : The straight line passing through P and dividing the area of the square by 1:3 does not exist. Can we know the locus of P and the area of the locus ? I ...
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6answers
659 views

538.com's Puzzle of the Overflowing Martini Glass - How to compute the minor and major axis of an elliptical cross-section of a cone

FiveThirtyEight.com Riddler Puzzle / May 13 The puzzle goes like this; "It’s Friday. You’ve kicked your feet up and have drunk enough of your martini that, when the conical glass (🍸) is upright, the ...
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2answers
88 views

How many mutually orthogonal circles are possible?

How many mutually orthogonal circles is it possible to have? It is easy to construct $3$ mutually orthogonal circles, e.g. $3$ circles with radius $1$ and centers at the vertices of an equilateral ...
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1answer
22 views

Find radius of a circle from intersecting chords

Say I have two chords that intersect inside a circle, not at a right angle, and neither is the diameter. It seems to me this is enough information that the circle must be unique, but I can't seem to ...
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1answer
23 views

Application of Rouche theorem in order to find the roots of a polynomial in each quadrant.

I want to solve the following : (i) Show that $z^4+2z^2-z+1$ has exactly one root per plane quadrant. My idea to prove (i) is by using Rouche theorem, by considering 4 cuts of the complex plane ...
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1answer
21 views

Problems on measure of angles and arcs in a circle diagram

A friend of mine recommended this site. I cannot figure out any of the parts in the problem in the picture click here The line segments AE and DE are not tangent to the circle, so I don't know where ...
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3answers
42 views

Same perimeter and area for a circle and an ellipse

For a given circle, is there exist an ellipse with same perimeter and area as to that circle? If not, that is my suspicion, is in three-dimension parallel question: For a given sphere, is there ...
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1answer
37 views

Geometric Optimization

In 1990 W. Kuperberg conjectured that it is impossible to have seven infinite mutually disjoint unit cylinders all touching a unit sphere. As a first step towards a solution I would like to answer the ...
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2answers
54 views

Rational Distance Problem triple — irrational point

Many points with rational coordinates are known with rational distances to three vertices of a unit square. For example, the following points are rational distances from $a=(0,0)$, $b=(1,0)$, and ...
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1answer
34 views

Formula for area of triangle in complex plane [closed]

If $A(z_1)$, $B(z_2)$, $C(z_3)$ are vertices of a triangle $ABC$ in Argand plane, what is the area of the triangle?
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2answers
27 views

The fact that surface of spherical segment only depends on its height follows from symplectic geometry

It is quite quite well known that the surface of the piece of a sphere with $z_0<z<z_1$ for some values of $z_0,z_1$ is given by $ S = 2\pi R (z_1-z_0) $. So this surface area only depends on ...
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3answers
76 views

There is a square $Q$ consisting of $(0,0), (2,0), (0,2), (2,2)$

There is a square $Q$ consisting of $(0,0), (2,0), (0,2), (2,2)$. A point $P$ satisfies following condition: The straight line passing through $P$ and dividing the area of square $Q$ in ...
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3answers
32 views

Calculate area of a figure on the picture

What kind of figure is it? (the filled one). How can I calculate it's area? Known: radius of each circle and coordinates of their centers. Picture of this figure
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0answers
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How to calculate the length of a variable radius helix between two points?

I have two points P1 = (R1,C1,Z1) P2 = (R2,C2,Z2) R: Radius C: Angle Z: Distance along axis of helix. If I linearly interpolate between these two ...
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1answer
36 views

Is point $A$ located in part of a circle in three dimensional space?

Given a point $A(x_a,y_a,z_a)$, a circle $O((x_O,y_O,z_O),r)$, and an circular sector $BOC$ constructed by two rays $\vec{OB}$ and $\vec{OC}$ starting from the center $(x_O,y_O,z_O)$ where the angle ...
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3answers
239 views

What is the moment of inertia of a Gosper island?

We know that regular hexagons can tile the plane but not in a self-similar fashion. However we can construct a fractal known as a Gosper island, that has the same area as the hexagon but has the ...
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0answers
19 views

Good way to plot coordinate system in computer?

I want to plot a coordinate system rotation in my paper, I want to know what would be a good way to make the plot? The plot would look like:
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3answers
41 views

How to graph $x^2 -4x$?

I know about transformations and how to graph a function like $f(x) = x^2 - 2$. We just shift the graph 2 units down. But in this case, there's an $-4x$ in which the $x$ complicated everything for me. ...
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2answers
159 views

The minimum value of $\frac{a^3 + b^3 + c^3 }{\sqrt{a^4 + b^4 + c^4 }}$ . When $a^2 + b^2 + c^2 = 1 $

Asume $a, b, c $ is non-negative real. I got above equation at this situation ; $\vec {x}= (a, b, c)$ , $\vec {y} = (a^2 , b^2 , c^2 ) $ $$ cos \phi = \frac{ \vec x \cdot \vec {y}} { \Vert {\vec ...
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3answers
70 views

Area of triangle determined by three vectors in $\mathbb{R}^3$

Given vectors $a = (2, 1, 3)$, $b = (4, 1, 2)$, $c = (1, -1, 5)$, need to find the area of the triangle $abc$ determined by the three vectors (the vectors are the vertices of the triangle). My ...
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1answer
61 views

Area of a rectangle.

What is the area of a rectangle that measures 4/10 x 4Ft. My grandson's teacher says the answer is squared = 1 2/5 squared
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1answer
46 views

Trigonometric number theory

Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + ...
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2answers
70 views

Where does this formula for the volume of a n-dimensional ball come from?

I recently came across the following formula for the volume of an n-dimensional unit ball: $$\frac{\pi^{n/2}}{\Gamma(n/2 + 1)}$$ Why exactly does this formula work?
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3answers
53 views

Find the shortest distance between the line $x + 2y = 1$ and the origin

The question is as stated, find the shortest distance between the line $x + 2y = 1$ and the origin in the coordinate system
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1answer
48 views

Counting Turns in a Rectilinear Spiral Graph

So consider a rectangular spiral graph which starts at the origin, goes right 1, up 1, left 2, down 2, right 3, ... (in units). How can we tell how many turns there have been given a point? For ...
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1answer
16 views

n-dim volume of a scaled ball

Let $B \subset \mathbb{R}^n$ be the unit ball with respect to an arbitrary norm $\|.\|$ (e.g. $B=\{x \in \mathbb{R}^n:\|x\| \le 1 \}$). I read in a book that it is easy to show: $vol_n(\epsilon ...
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0answers
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How to transform set points in a any polytope

I need transform a any set point in a any regular polytope, e.g., twenty points in 5-simplex. Is possible? Thank you so much.
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What are the exact angles of the triangular faces of each of the rhombic pyramids of the icosahedron stellation, the compound of five octahedra?

I'm talking about this compound of five regular octahedra. Based on looking at the icosahedron stellation diagram and making an educated guess about where the golden ratio comes in, I calculated the ...
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1answer
41 views

Maximun no. of diagonals that can be drawn so that all the parts they divide into are triangles?

In a convex n-gon (n>4) no three diagonals are concurrent (intersect at the same point). What is the maximum number of the diagonals that can be drawn into this polygon so that all the parts they ...
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2answers
766 views

Modelling the “Moving Sofa”

I believe that many of you know about the moving sofa problem; if not you can find the description of the problem here. In this question I am going to rotate the L shaped hall instead of moving a ...
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4answers
76 views

Construct parallel through a triangle satisfying a sum condition

I would like to draw, using the classical compass and rule methods, some points $D$ and $E$ given a triangle $\Delta ABC$ such that $BD + EC = DE$ and $DE$ is parallel to $BC$, as in the following ...
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constrain general solution of ill-posed linear system to $\Re_+$?

I have a solution space of an under-determined linear system Ax = b with n x m matrix A: $$x= x0 + V2 * c (1)$$ with [U, S, V] = svd(A); V2 = V(:,r+1:end); $x0 = A^+ b; $ r = rank(A); I ...
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Can I always find 2 independent axises from data distributed in x, y panel?

Let's say, I have data (x, y) distributed at x,y panel: Can I always find a pair of (u, v) axis, so that data along these axis are independent from each other? u, v are just x, y rotate at ...