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

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2
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1answer
572 views

Distance between point and plane & orthogonal projection matrix

I am poor in mathematics and want to learn few fundamental ethics to understand some of advanced things; For plane $i$, denote $n_i\in\mathbb{R}^3$ and $o_i\in\mathbb{R}^3$ respectively as its normal ...
1
vote
0answers
121 views

Quaternion barycentric interpolation

Let's say that i have a set of quaternions, each representing a 3-angle orientation. And with each quaternion is associated a real value (let's say a speed value for explanation's sake). Now with an ...
1
vote
2answers
99 views

Any four points on the space curve given by the parametrization $(t,t^2,t^3)$ are noncolinear

I want to draw a space curve given by the parametrization $(t,t^2,t^3)$, Thanks to wolfram, i have an idea how it looks like but is there any way to draw this? and I want to show that any four points ...
4
votes
1answer
233 views

Maximum number of points in unit cube

What is the maximum number of points that can be within a unit cube (no points on cube vertices, faces, or edges) such that no two points are within 1 of each other? I'm asking because I'm creating a ...
8
votes
4answers
968 views

Is there any good reason why a protractor starts from right to left, unlike a scale, which starts from left to right?

While studying through the number system, i notice that positive side is from 0 to +ve infinity. The direction is left to right. However, this is opposite in case of angles. The sort of curved number ...
2
votes
1answer
47 views

Area of Questionably Generated Manifold

I might not possess the language to ask this question, but I'm going to try anyway. Consider a path c(t) : $\mathbb{R}\rightarrow \mathbb{R}^n$. Let c'(t) denote the tangent vector of the path c(t). ...
11
votes
11answers
1k views

Text suggestion for linear algebra and geometry

I want to study more linear algebra over the summer, specifically relating it to geometry. I was originally going to read Shafarevich's Linear Algebra & Geometry, after a recommendation, but it ...
5
votes
2answers
399 views

Has anyone discovered a convex space-filling 15-faced polyhedron?

I've been looking for extensive surveys regarding space-filling polyhedra, but have only come across Michael Goldbergs "Convex polyhedral space-fillers of more than twelve faces" from 1979, stating ...
4
votes
1answer
205 views

Geometric Inequality Related To Median, Altitude

For a triangle $ABC$, let $m_{a}$, $h_{a}$ be $A$-median, $A$-altitude. Define $m_{b}$,$h_{b}$ and $m_{c}$,$h_{c}$ likewise. Prove that $\dfrac{h_{a}}{m_{b}}+\dfrac{h_{b}}{m_{c}}+\dfrac{h_{c}}{m_{...
2
votes
1answer
39 views

Can this logic about locating a point uniquely without using “-” be challenged?

I was just exploring a possibility of locating point without using "-" ( negative ) sign. ( Actually negative sign confuses me a lot when understanding the basics of coordinate geometry ) So, here ...
3
votes
1answer
136 views

Group acting by isometries on a length space

I am reading the book A course in metric geometry by Burago, Burago and Ivanov. I have some difficulties with an exercise 3.4.6 on page 78. The exercise is the following: Let a group $G$ act by ...
3
votes
1answer
192 views

Questions about flowing curves

A closed smooth non-self-intersecting curve in $\mathbb{R}^2$ having its curvature less than one is called a flowing curve. The three connected questions arise: How to prove that a disk of radius 2....
3
votes
5answers
958 views

Higher Dimenional Tic Tac Toe

Here we have a problem that seems very intuitive, but is hard to define mathematically. In Tic Tac Toe, we can find an equivalent of the game in any number of dimensions, it seems. The trick is to ...
4
votes
1answer
191 views

Calculate polyhedra vertices based on faces

I have some origami polyhedra which I know the type of faces it has and how they are connected (such as this torus) and I want to calculate the co-ordinates of the vertices to use as an input to ...
0
votes
1answer
375 views

Number of sides a regular polygon has.

The question is "Both tile A and B are regular polygons. Work out the number of sides A has." For this I put B is equilateral ∴ all angles are 60. However, I have no idea where to go from ...
3
votes
2answers
3k views

Deriving the formula for the volume of a sphere

A circle $x^2 +y^2 =a^2$ is rotated about the $y$-axis to form a solid sphere of radius $a$. How do you express this motion mathematically in such a way that it allows me to arrive at the formula ...
1
vote
3answers
2k views

Triangle inscribed in circle, vertex at circle's center, solve for unknown angles.

$O$ is the center of the circle , $A$ and $B$ lie on the circle what are the possible values of $x$ and $y$ I found answers options , asked to mark one or more ...
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vote
3answers
112 views

Find coordinates of vertex of equilateral triangle

$ABC$ is an equilateral triangle , $AC = 2 $ What is the value of $p$ and $q$ ?
0
votes
2answers
35 views

What is the ratio of the perimeter of $OPRQ$ to the perimeter of $OPSQ$?

Area of circle $O$ is $64\pi$. What is ratio of the perimeter of $OPRQ$ to that of $OPSQ$ ($\pi = 3$)? Okay i have tried couple of things but seems its not working . Please suggest me proper ...
2
votes
3answers
41 views

How to solve part II of this question?

Given the points $A(4, 2)$ and $B(1, 1)$, find the equation of the line $l_1$ perpendicular to $AB$ passing through the point $B$. The point $C$ lies on $l_1$ such that $ABC$ is an isosceles ...
1
vote
1answer
84 views

Two dimensions geometry problem

Determine the vector equation and parametric equations of the line in $\Bbb R^2$ passing through $A=(1, 3)$ parallel to the line passing through $B=(-1, 4)$ and $C=(2, 1)$. Find out if the line ...
19
votes
1answer
979 views

Are there prime lengths in triangle with all integer sides and heights?

Suppose you have a triangle in which all sides and all heights are integer in length (i.e. triangle with sides 20, 25, 15 has heights 15, 12 and 20). Could it be that at least one of those numbers is ...
1
vote
0answers
37 views

generalizing symmetry axis of elliptical contours in 3D

The contours of the following function $f$ trace out an ellipse, $f(x, y, z) = \exp(-x^2a)\exp(-y^2b)$, where $a\neq b$ are positive, real constants greater than zero. The axis of these ellipses is ...
4
votes
3answers
131 views

Points on a sphere

We draw n points, A, B, C, ... Z, on the upper hemisphere of a sphere, and their n antipodal points on the lower hemisphere, a, b, c, ..., z. We draw the n(n-1)/2 great circles connecting each pair ...
9
votes
1answer
338 views

Pigeonhole principle for a triangle

Consider a equilateral triangle of total area 1. Suppose 7 points are chosen inside. Show that some 3 points form a triangle of area $\leq\frac 14$.
1
vote
1answer
148 views

Line segment, sum of supplementary angles to internal ones.

Having hard time understanding this drawing. Essentially the questions about finding the sum of the supplementary angles to the internal angles of a non standard polygon. I have the answer as roughly ...
1
vote
1answer
94 views

How to prove this fundamental relationship $ b=\ell+n-1$?

How to prove this fundamental relationship? In a network or circuit, number of loop, nodes and branches has to satisfy the following fundamental relationship: $$ b=\ell+n-1,$$ ...
2
votes
1answer
336 views

The angle at which a circle and a hyperbola intersect?

$x^2 - 2y^2 = 4$ $ (x-3)^2 + y^2 = 25 $ How do you calculate the angle at which a circle and a hyperbola intersect? If I express $y^2$ from the first equation and apply it to the second equation, ...
0
votes
1answer
491 views

Goldberg polyhedra coordinates

I would 3D-print some Goldberg Polyhedra importing in Sketchup, the coordinates provided on these links: 72 faces (2,1) - (coordinates) 132 faces (3,1) - (coordinates) 192 faces (3,2) - (coordinates)...
7
votes
1answer
91 views

A simple question on simplex

Let $S \subset \mathbb{R}^n$ be a $n$-simplex. Let $a_0,\dots, a_n$ be the vertices of $S$. Define $L_i\subset \mathbb{R}^n$ be the hyperplane which touches $S$ at $a_i$ and parallel to the convex ...
0
votes
1answer
194 views

Check for k convex polygon

Given a list of lengths, how can I check whether it will form a $k$ convex polygon? For example, given $k=3$, and lengths $1,1,1$, the answer is yes.
0
votes
5answers
790 views

Internal/external division of a line question with unknown coordinates?

I've been having trouble with this question: "Given $K(3, - 1)$ and $L(-4, 2)$, find two positions of $A$ on $KL$ such that $KA = 2( KL)$." Any help would be appreciated. The first thing I tried was ...
3
votes
1answer
148 views

finding the maximum area of 2 circles

An equilateral triangle with height $h$ has 2 different incircles. the bottom circle is tangent to the base of the triangle at the middle point of the base. what should be the radius of the upper ...
5
votes
3answers
4k views

An equation about a rectangle with given perimeter

I am doing a revision calculator paper and am stuck on an algebra question. There is a picture of a rectangle. One side is $x-2,$ another side is $2x +1.$ The question is. Setup and solve an ...
5
votes
3answers
403 views

If $a$ and $b$ are non-negative real numbers, prove that $ab(a+b) \leq a^2+b^2$.

If $a$ and $b$ are non-negative real numbers, prove that $ab(a+b) \leq a^2+b^2$. Is is a geometric mean? How to prove it?
6
votes
4answers
753 views

Confusion about the usage of points vs. vectors

As far as definitions go, understand the difference between a vector and a point. A vector can be translated and still be the same vector, whereas a point is fixed. But I would like some clarification ...
0
votes
2answers
98 views

Coordinate geometry and translations: rotations and composites oh my!

Here's the problem: Let $R_y$ be a reflection in the $y$-axis and $T : (x,y) \rightarrow (x-3,y+1)$. Which one of the following transformations is equivalent to $R_y \circ T$? Here's my thought ...
1
vote
0answers
63 views

Meaningful measures for comparing infinite dimensional geometric objects

I have two infinite-dimensional convex polytopes, call them $A$ and $B$. I know that $B$ is completely contained within $A$, and I want to say something meaningful about their relative sizes. From ...
0
votes
1answer
32 views

implement a linear gradient on a tube

I have the following function, which traces out a circle for a certain value of $f$, $f(x, y, z) = \exp(-x^2)\exp(-z^2)$. For a given $f$, the circle has the same value for all $y$ in my coordinate ...
0
votes
1answer
189 views

proof of ratio of area of quadrilaterals

In the convex quadrilateral ABCD, points M and N lie on AB such that AM =MN=NB.Points P and Q lie on side CD such that CP=PQ=QD.How can we prove that area of AMCP is 1/3 of ABCD?
3
votes
1answer
2k views

Polygon Inequality

We know that to form a triangle the 3 sides should obey the triangle inequality . So is there any rule to be followed by the sides of $n$-sided convex polygon. For Eg:- $1,2,4$ cannot form a triangle ...
4
votes
1answer
177 views

Strong equidistribution of points on the n-sphere

The vertices of a Platonic solid are equally distributed on its circumscribing sphere in a very strong sense: each of them has the same number of nearest neighbours and all distances between nearest ...
3
votes
1answer
476 views

Do the tangents of two circles define concentric circles?

Given two non-overlapping circles, $R_1$ and $R_2$. The radii of $R_1$ and $R_2$ may be different. The distance between the centers of $R_1$ and $R_2$ is defined as $x$. Draw the four tangents ...
2
votes
1answer
781 views

Angle between different rays (3d line segments) and computing their angular relationships

I have several positions (say A,B,C,..) and I know their coordinates (3d). From each point, if a certain ray is passing in a way to converge them at a given (known) point (say O). This point O is ...
0
votes
1answer
1k views

Image under a translation -> preimage

I have the following problem in coordinate geometry. Note that Z simply refers to a point. The image of $Z(3,-8)$ under a translation is $Z'(2,2)$. What is the preimage of Z? I have no idea how ...
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vote
2answers
186 views

Simple question about circumference of circle

Q: The physical education teacher asked to one classroom, by vote, choose a sport between volleyball, basketball and football, to practice in class the following week. pie chart: The segment AB, ...
0
votes
3answers
251 views

Goat tethered in a circular pen

There is a circular pen with a goat in it. The goat is tethered by a rope to the edge of the pen. The rope is the length of the radius of the pen. What area of grass in the pen can the goat graze?
0
votes
1answer
61 views

Generate a Normal in 3D Without Branching?

I have a vector $v$ in arbitrary 3D space ending at point B. In order to generate the next point -- C, I uniformly pick an ...
2
votes
1answer
114 views

A Fuchsian Group?

Let $p_k := e^{\pi/2 i k}$, $k \in \{0, 1,2,3\}$. Let $b_k$ the geodesic of the hyperbolic disk connecting $p_k$ and $p_{k+1(\text{mod}4)}$. For instance, $p_0$ and $p_1$ are connected by the lower ...
1
vote
1answer
8k views

How many possible line segments can be found in a collinear line with an x number of points?

What would be the method or the formula for this kinds of problems/equations? A given example would be: How many possible line segments can be formed in a collinear line with 6 points? What is the ...