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

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Number of Rotations of a unit cube

Let $C $ be the unit cube $[-1,1]^3 \subseteq \mathbb R^3$.How many rotations are there in $\mathbb R^3$ which take $\mathbb C$ to itself? Please help me to visualize this.
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Locus given by a pair of scissors sliding along the ground.

I came up with this problem when dragging a pair of scissors along the ground. The question is, more mathematically: Suppose there is a point (a,0) and a point (0,b) with a fixed distance m between ...
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0answers
8 views

Discrete set of points $\xrightarrow{to}$ Curve $=$ Parameterization?

I have to solve a problem about generating 2D vector images from a collection of 2D points that shape a 2D figure, and also I should algorithmically find a 3D NURBS/set of curves starting for a ...
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1answer
32 views

Geometric meaning of $z=\frac{x_1+x_2}{2}$, $y=\frac{x_1+x_2}{|x_1+x_2|}$, and of a substitution (in the complex plane)

Let $S$ be a unitary (that is, $r=1$) circle centered in the origin of the complex plane. Let $x_1 \in S$ and $x_2 \in S$ be the vertices of the triangle $Ox_1x_2$. What is the geometric meaning of ...
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1answer
20 views

Intersections in polygons

I'm having troubles solving the following problem which is about combinatorics : let n a natural number >= 3, and a convex polygon with n vertices. Each vertices are supposed to connect each other ...
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20 views

The line of two conjugates poles make a harmonic set

Let $A,B$ be two conjugate points respect to a circle $K$ of center $O$ and radius $k$ and let $C,D$ be the intersection points of the line $AB$ and the circle $K$. (A and B are conjugates if the ...
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1answer
14 views

Projected Area vs. Surface Area of a 3D Set

(In what follows, I'm making up the nomenclature as I go along, so please pardon anything nonstandard.) Suppose I have a set of points $A \in \mathcal{R}^3$ which is compact, convex, and simply ...
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1answer
14 views

Ray-sphere intersection: t-value of the intersection points

You have a sphere centered at [1,2,3] with radius 3, and a ray from [10,10,10] in the direction [-1,-1,-1]. Write the implicit equation for the sphere, the parametric equation for the ray, and compute ...
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31 views

Limits, tangents and areas. Why are these statements intuitive?

I'm reading Calculus: Early Transcendentals, by Anton, Bivens and Davis (9th edition). I'm not understanding a few things. Could someone please help me? On page 68 it says: "suppose that we are ...
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1answer
30 views

Prove that $ABC$ is equilateral

Let $D,E,F$ be points on the sides BC,CA,AB respectively of a triangle $ABC$ such that $BD=CE=AF$ and $\angle BDF=\angle CED=\angle AFE$.Prove that $\triangle ABC$ is equilateral. My attempt - Using ...
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6 views

matrix of finsler metric is a positive semidefinite

Suppose $M$ be a smooth manifold, a map $F:TM \to [0,\infty]$ is called a finsler metric on $M$ if $F_{x}(y) = F(x,y)$ be a minkowski norm on $T_{x}M$ for every $x\in M$. A symmetric matrix $A\in ...
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20 views

Prove $OD$ is the angle bisector of the angle BOC

Let $ABC$ be a non-isosceles triangle and $I$ be the intersection of the three internal angle bisectors. Let $D$ be a point of BC such that $ID\perp BC$ and $O$ be a point on AD such that $IO\perp A$D ...
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1answer
43 views

Largest regular hexagon inside a square [on hold]

What is the side of the largest regular hexagon which can be drawn inside a square of side $x$?
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14 views

Existence of half-planes with respect to regular open sets of the Euclidean plane

Let $\langle\mathrm{r}\mathscr{O},\mathord{\subseteq}\rangle$ be the complete Boolean algebra of open domains (regular open sets, these that are equal to the interior of their closure: ...
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1answer
23 views

Area of square created by intersection of segments from a square vertexes to their opposite sides

There will be an square created when we draw segments from a square vertexes to their opposite sides' middle. What is the relation between smaller square's area and the side length of the bigger ...
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0answers
13 views

What is the name for the class of n-dimensional areas which can be described by 2 points

I wonder if there is a name for the class of n-dimensional areas which can be described by 2 points. Examples: 1-dimensional: line (startpoint,endpoint) 2-dimensional: rectangle (2 opposite corners) ...
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1answer
20 views

Equation of a specific shape's edge?

Suppose we have such a shape: It is needed to found what this shape's edges are. I mean, this shape edges are: outer arc (upper) - we know everything we would like about this arc: radius, start ...
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3answers
14 views

Cyclic quadrilaterals - finding the size of an angle

I know this might seem like a really simple question, but I really don't understand where I am going wrong. I am familiar with cyclic quadrilaterals as well as their properties, but this question ...
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0answers
14 views

Cover points with specified amount of cuboids and minimize overlap

Given a list of Points (the coords are pure integeres), I want to cover all of them with cuboids. The Problem is, I have a limited number of cuboids I can use. Of course I would like to have a ...
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1answer
17 views

Area of a triangle.

The area of a triangle $ABC$ is $144$.Denote the midpoint of $BC$ by $P$,of $AP$ by $Q$ and of $AC$ by $R$.Calculate the area of the triangle $PQR$. I draw the picture but I do not have any idea to ...
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21 views

Calculate the distance between any points in two different circles

I have two overlapping circles (C1 and C2) for which the distance between their centers is know. Inside each circle theres's random number of points (P11... P1n and P21... P2n) for which the distance ...
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1answer
23 views

Estimating the missing points of a 3D point cloud

Consider a cloud of N points (forming a smooth 3D object), in which n points are missing. Also, consider that there is no prior knowledge about the original shape of the point cloud. The only ...
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1answer
15 views

Bi-conditional statement

Can a bi-conditional be written with these 2 statements? If 3 points are collinear then they are coplanar. If 3 points are coplanar then they are collinear.
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1answer
43 views

Moving a point around a circle

we're currently working on a game which involves a character that rotates around a point. We are using a rotation matrix to rotate a given a point (x,y) around another point by first translating to ...
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1answer
33 views

Dividing a disc into equal parts

Prove that it is not possible to divide a disc into $7$ parts of equal area by means of three straight lines. Background: I saw this question asked in a way which seemed to imply the possibility ...
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1answer
15 views

If two harmonic quartets have a common point, prove their lines are concurrent

Let $A,B,C,D$ and $A,L,M,N$ be collinear points such that $\{AB,CD\} = \{AL,MN\} = -1$. Prove that the lines BL, CN and DM concur. I tried to build a triangle using A as a common point and then use ...
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1answer
25 views

How to work out side length of a square with 3 unit circles

How do you work out the side length of a square which contains 3 packed circles of radius 1: "Circles packed in square 3" by Toby Hudson - Own work. Licensed under Creative Commons ...
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0answers
15 views

Find all regions formed by a set of circles

I was doodling with Python to draw some circles, and I was able to find all intersection points of a set of random circles, yay ! Now I'm stuck on a question, is there a way to find all regions ...
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0answers
35 views

Area of octagon constructed in rectangle

Question If each vertex of a rectangle connected to midpoint of to not adjacent sides then a octagon will be formed find the ratio of octagon area to rectangle area Things that can't be ...
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1answer
37 views

the set of points equidistant from $ u $ and $v$ form a line.

Let $u$ and $v$ be two vectors in $ \mathbb{R}^2 $ with the standard norm. Show that the set of points equidistant from $ u $ and $v$ form a line. I show that if $x$ is equidistant from $u$ and $v$, ...
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0answers
21 views

Determine sine wave frequency from two arbitrary points

If I have only two arbitrary points on a sine wave, what would be the simplest method for determining the frequency of the sine wave? The frequency is unknown. The bandwidth is restricted, the time ...
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3answers
14 views

Find the radii of the two circles which pass through the point $(16,2)$ and touch both axes

How can I find the radii of the two circles which pass through the point $(16,2)$ and touch both the axes? I've only ever seen demonstrations using three normal co-ordinates; or two normal ...
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1answer
20 views

Find Equation of Parabola

I am trying to get my head around parabolas and running into bit of a wall. I've been trying to figure out what the formula for a parabola would be given that i have 2 points on Cartesian plane ...
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1answer
13 views

Prove that Lorenz's Postulate is logically equivalent to Parallel Postulate 5

Lorenz, Every single line through a point within an angle will meet at least one side of the angle. I know I have to Show that the parallel postulate 5 implies lorenz, and then lorenz implies ...
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1answer
30 views

Find the missing angle of similar triangle

Find the missing angle $\theta$ in the triangle below given that $R>r$, $l\geq R$, $0< \theta < \frac{\pi}{2}$. Attempted Solution I attempted to use similar triangles to find the angle ...
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2answers
28 views

Algebra Logical Pythagorean theorem help

A wire is attached to the top of a pole. The pole is 2 feet shorter than the wire, and the distance from the wire on the ground to the bottom of the pole is 9 feet less than the length of the wire. ...
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2answers
26 views

How to check if a 3D line segment intersects a cylinder?

I have developed a check for a 2D case of a circle intersecting a 2D line segment, however there is a particular case that I can't figure out how to extend to 3D: If one endpoint on the 3D line ...
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1answer
25 views

Finding the intersection points of two circles algebraically

I need to find the intersection points of two circles with equations: $(x+1)^2+(y-1)^2=1$ and $(x-1)^2+(y+1)^2=4$. I understand how to find the points by plotting the circles but I am unsure of how ...
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29 views

Largest $n$-vertex polyhedron that fits into a unit sphere

In two dimensions, it is not hard to see that the $n$-vertex polygon of maximum area that fits into a unit circle is the regular $n$-gon whose vertices lie on the circle: For any other vertex ...
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0answers
11 views

Applying homography to ellipse derived from normal distribution

I need to apply a homography to an elliptic area. First question: Is the resulting also elliptic in every case? I think so, but actually i don't really know. Anyway, I assume it for this question. ...
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0answers
17 views

Right angle triangle with integral vertices [duplicate]

There is right angled triangle with known lengths for each side. Is there a known method to check if all the vertices will be integers if that triangle is placed anywhere on the plane? (or) what is ...
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2answers
126 views

Need some help solving high-school level trignometry question.

here it is. I've tried solving it multiple ways but it gets too complicated. Is there any way to solve this?
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1answer
18 views

Observation about SSA criteria.

IS the following observation true in general? I nned some help please.
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1answer
11 views

Analyze growth using graphs

How do I analyze growth using only graphs (and no exact values)? I want to show that the first growth is cubic, the second is square and the third is linear. As far as I know the third one can be ...
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2answers
64 views

Is HHH a congurence criteria for triangles?

I wanted to know if a triangle defined by its 3 heights is unique. I took this up as a challenge but was able to get nowhere, can anyone help me? :)
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1answer
28 views

When is $3R\le 2h_{\max}$ true for acute triangles?

I was working on a problem recently, and it happened that it could be solved if $3R\le 2h_{\max}$ was true for all acute angled triangles. So I used GeoGebra to check it, and found that for some ...
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3answers
131 views

Drawing a Right Triangle With Legs Not Parallel to x/y Axes?

I have been presented with an interesting problem. How can I decide whether a right triangle with given side lengths can be placed (with integer coordinate vertices) on a Cartesian plane so that the ...
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2answers
24 views

Proove the following using either Direct Proof, Contrapostive and Contradiction. (Question related to Geometry).

A circle has centre $(2,4)$. Prove that if $(0,3)$ is not inside the circle, then $(3,1)$ is not inside the circle. I just want to know if my method would be correct. The method I used is as follows: ...
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1answer
29 views

The reflection of $f(x,y) = x^2 - y^2$

How would I make a reflection of $$ f(x,y) = x^2 - y^2 $$ along the z axis? Beacuse if if write $$ f(x,y) = -(x^2 - y^2) $$, flips the figure along the XY axis...
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1answer
17 views

How many milliliters of liquid to fill [duplicate]

A right circular cone has a depth of 103 mm and a top diameter of 82.4 mm. The cone contains water to a depth of 30.0 mm. How many more millilitres of liquid need to be added in order to fill the ...