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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.

5
votes
1answer
50 views

What is the average distance of point in hypercube to its center?

How do I compute the average distance of point inside an hypercube to the center of the hypercube as a function of the dimensionality of the space? Here I consider the hypercube defined as $C_n=\{x\...
0
votes
1answer
17 views

Convert point from a plane to another

I have two planes: Plane A $[x,y]$ Plane B $[x,y]$ $[0,0]$ of $A$ is on $[0,0]$ of $B$. However, the axis $x$ of plan A doesn't have the same angle than the one of B. I have the angle of B, and the ...
3
votes
1answer
49 views

Eccentricity of conic given by a complicated equation with trigonometric coefficients such as $\tan 10^\circ$

Find the eccentricity of the conic given by: $$\left(x\tan 10^\circ+y\tan 20^\circ+\tan 30^\circ\right)\left(x\tan 120^\circ+y\tan 220^\circ+\tan 320^\circ\right)+2018=0$$ What I have tried $$\...
2
votes
2answers
78 views

In $\Delta$ABC, $AB=5$ and $E,F$ are two points on $BC$ such that $BE=1,EF=3,CF=2. $

In $\Delta$ABC, $AB=5$ and $E,F$ are two points on $BC$ such that $BE=1,EF=3,CF=2. AE$ and $AF$ intersect the circumcircle of $\Delta$$ABC$ at the point $G$ and $H$ respectively. $GH$ and $BC$ are ...
3
votes
3answers
692 views

How many of these lines lie entirely in the interior of the original cube? [on hold]

A portion of a wooden cube is sawed off at each vertex so that a small equilateral triangle is formed at each corner with vertices on the edges of the cube. The $24$ vertices of the new object are all ...
0
votes
1answer
59 views

value of $a+\frac{1}{b^2}$ in straight line

The sides of a triangle have the combined equation $x^2-3y^2-2xy+8y-4=0.$ The third side, which is variable always passes through the point $(-5,-1)$ . If the range of values of slope of the third ...
0
votes
0answers
91 views

Two definitions of $\pi$

I have the feeling that ancient mathematicians (like Greek or Chinese), trying to find good approximations of $\pi$ used two definitions: If $A$ is the area of a disk and $r$ is its radius, $\pi=A/r^...
0
votes
1answer
18 views

Find ratio of the volume of two cone

Given two sector ABC and PQR, $\angle A=2\theta$, $\angle P=3\theta, AC=2r, PR=3r, $ both sectors are folded into a right circular cone, find the ratio of the volume of two cone. I am having ...
0
votes
2answers
75 views

How to use an iterative method to compute y values corresponding to an x value of a rotated ellipse?

I am trying to render the outline of a rotated ellipse on raster graphics. The general form of a conic is: H(x, y) = A x² + B xy + C y² + D x + E y + F = 0 I am currently able to render the ellipse ...
1
vote
2answers
23 views

Formula (how to calculate) Y axis cross-point of two intersecting lines

i.e. I have two lines: A) Orange (Y axis starts at: 6, end at: -3) B) Green (...
0
votes
0answers
28 views

Discrete Geometry Pre-requisites

I have started studying Lectures on Discrete Geometry by Jiri Matousek, chapter 1 was fine for me I am midway chapter 2 but I am not understanding most of things after this. Are there any other books ...
0
votes
0answers
10 views

Parametric and implicit equations of $k$-flats (lines, planes, etc)

We know that any $k$-flat in $\mathbb{R}^n$, $k<n$, can be expressed either as a system of $m$ equations with $k=n-m$ and $$0 = c_{i,0}+\sum_{1\leq j\leq n} c_{i,j} x_j \quad ,i=1\dots m$$ or as $n$...
0
votes
0answers
40 views

Perspective Puzzlery! Determine Distance and Angle of View Point, from a Rectangle, in a Digital Image.

Given points; A - (341, 81) B - (265, 630) C - (884, 459) D - (896, 942) represent the coordinates of the corners of a rectangle in a digital photo. The true distance of Line AB=CD is 24 ...
2
votes
0answers
45 views

In $\triangle ABC$, if $3a=b+c$, then what is $\cos\frac{B}{2}\cdot\cot\frac{C}{2}$?

In $\triangle ABC$, we have that $3a=b+c$. Then, what's the value of $$\cos\frac{B}{2}\cdot\cot\frac{C}{2}$$ First I used auxiliary formulae of $\cos B/2$ with $3a=b+c\implies s=2a$, but I am ...
-2
votes
1answer
27 views

How to find what degree on a circle is tangent to a point outside of that circle?

I know the (x,y) of a point, P, outside of a circle. I know the (x,y) for the origin of a circle, O. I know the radius, r, of that circle. How would I find what degree (e.g. 20 degrees, 270 degrees) ...
0
votes
0answers
37 views

Changing rotation center

Things that we have: 2 dimensions, a object with it's coordinates (object P1), it's rotation center (pivot) C1. After that lets rotate it at pivot C1 by known angle A. Now let's move that pivot by ...
6
votes
2answers
98 views

Proof that there exist only 17 Wallpaper Groups (Tilings of the plane)

I had a professor who once introduced us to Wallpaper Groups. There are many references that exist to understand what they are (example Wiki, Wallpaper group). The punchline is $$There \,\, are \,...
1
vote
1answer
39 views

Special case for Giraud's Theorem

I was wondering how Giraud's Theorem would work for spherical polygons. Do we know a proof?
2
votes
1answer
96 views

For $P$ an arbitrary point in $\triangle ABC$, show that $\sum_{cyc}c(\sin \angle CAP+\sin\angle CBP)\leq a+b+c$

In the interior of $\triangle ABC$ we take the arbitrary point $P$. Prove that the following inequality holds: $$\small c(\sin\angle CAP + \sin\angle CBP) + a(\sin\angle ABP +\sin\angle ACP) + b(\sin\...
-1
votes
0answers
60 views

$RBC$ and $RAB$ have the same measure [on hold]

Let $ABC$ be a triangle with right angle $A$. Let $AA' \perp BC$ and $C'$ the middle of $(AB)$. Let $P$ the middle of $AA'$, $Q$ the intersection of $CC'$ and $BP$, $R$ the ...
0
votes
0answers
11 views

Distance to median vs average intra-distances

Consider $n$ points in a vector space, denoted $(a_1, \dotsc, a_n)$. I am wondering if the following inequality holds true: $$ \min_x \sum_{i=1}^n d(a_i, x) \leq \frac{1}{n-1} \sum_{i=1}^n \sum_{j \...
0
votes
0answers
15 views

Computational geometry relationship between 2 arcs

I'm writing a program which is making offsets for provided shapes. On the attached picture you can see example of my arc object and all known values. Let's assume that a direction is CW. $O$ - ...
2
votes
1answer
98 views
+50

Special proof for Pick's Theorem?

This is kind of a two-part question and I think one should lead to the other- Could someone show or point me to a proof of the theorem for primitive triangles. Also, let's say we have a lattice ...
3
votes
1answer
47 views

Square with the minimum possible area that satisfy the given conditions

We're given a square with integer side lenght, it is divided into 4 triangles such that all the triangle's side lenghts are integers and that no two triangles are congruent, what is the minimum ...
2
votes
2answers
38 views

Applying Pick's Theorem in the complex plane

Let's say that $X$ is a parallelogram with vertices that have integer coordinates, how could I prove that $X$'s area is an integer? I have to try working in the complex plane to so that the vertices ...
4
votes
2answers
49 views

Area of triangle in $\mathbb R^3$ given $3$ coordinates

I know that if we have $3$ points $a$, $b$,and $c$ in $\mathbb R^3$, the area of the triangle is given by: $\frac{1}{2}\|\vec{ab}\times \vec{ac}\|$. This means that the area of the triangle equals ...
0
votes
1answer
22 views

Calculating a point in 3D using the Pythagorean theorem

I'm trying to find out if it's possible to calculate Az If I'm just given Ax and Ay. As an example say Ax is 3.4 and Ay is 7.8 how can I go about finding Az? If your wondering I was watching a video ...
2
votes
0answers
50 views

Cutting a Solid Torus with $n$ Planes

Question: How many pieces a solid torus be cut into with three (affine) planar cuts? A google search will quickly reveal that the answer is thirteen, as can be read about here. The picture below ...
0
votes
0answers
34 views

Solving a area problem in a different way gives different solutions

today I was confronted with this little area problem from this video: https://www.youtube.com/watch?v=BgrWHOocYZA So I gave it a try and came up with a solution. But after comparing the solution with ...
-1
votes
1answer
39 views

What is the radius of the semicircle?

Diagram : a 40cm wire bent to make a closed figure that consists a rectangle and a semicircle. (Kinda like a door with a curved top) Total perimeter of diagram / length of bent wire (rectangle + ...
-4
votes
0answers
7 views

straight lines finding sum of two distance and their differences max or min [closed]

consider points A(1,2) and B(1,3) and M on line x+y=0 What should be M for: I) AM + BM TO BE MINIMUM II) AM-BM MAX III) AM-BM IS MIN I dont understand when these are minimum or maximum
0
votes
1answer
18 views

value of $m$ and $c$ in Circles

If line $y=mx+c$ is a common tangents to the given circles and $r_{4}=r_{1}+r_{2}$ and $r_{5}=r_{2}+r_{3}$ where $r_{i}$ is the radius of circle $C_{i}$ for $i=1,2,3,4,5$ Then value of $m+c$ is ...
1
vote
2answers
22 views

locus of point of intersection of family of lines

Consider the family of lines $(x-y-6)+\lambda(2x+y+3)=0$ and $(x+2y+4)+\mu(3x-2y-4)=0.$ If the lines of these two families are at right angle to each other. Then find the locus of point of ...
0
votes
1answer
32 views

Why are lengths different when an object is curved?

In the above diagram ad is equal to bc. Likewise, ce should be equal to df (Since angle F and E are at 90 degree). My question is why when calculating length of ce and df using $2\pi r$ they are ...
2
votes
0answers
40 views

When do three disks in the plane intersect?

Suppose $ABC$ is a triangle with $|AB|=c$, $|BC|=a$, $|CA|=b$. Suppose further that $A,B,C$ are the centers of three disks with radii $r_A,r_B,r_C$, respectively. Is there a sensible algebraic ...
3
votes
1answer
44 views

Characterizing triangles with integer-degree angles that are similar to any of their iterated orthic triangles

A triangle has integer angles $a,b,c$ (in degrees) and we create the triangle's pedal triangle and call it Pedal-$(1)$. (Here, "pedal triangle" means the specific pedal triangle whose vertices are the ...
0
votes
1answer
52 views

Using Ptolemy's Theorem to find length ratio

In this figure, $X, Y$ are tangent points and $\frac{DX}{EX} = \frac{8}{3} , \frac{EY}{DY} = 4 , \frac{AC}{AB} = \frac{5 }{4} . $ Then, what is $ \frac{BC}{AX}$ ? System of equations from the ...
1
vote
1answer
75 views

Is $[0,1) \cup \{2\}$ a manifold with boundary? My issue is the $2$.

This has been asked about here: Understanding topological and manifold boundaries on the real line, and Sharkos said Personally I'd say $M$ wasn't a valid manifold with boundary because the $\{2\}$ ...
1
vote
2answers
79 views

Foundational Areas of Math?

I am currently an undergrad student and recently changed my major to mathematics (after taking my first proof based math course). Unfortunately, now that I am taking several upper division math ...
0
votes
1answer
24 views

Difference between Orthic triangle and Pedal triangle

Is there any difference between Orthic triangle and Pedal triangle or both both are same ?
1
vote
0answers
18 views

Does classification of 1-manifolds with boundary give induced orientation of image of closed interval under a smooth immersion?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. I have been trying to wrap my head around this for about 2 hours (...
2
votes
0answers
62 views
+50

Using Coordinate Descent on Projected Space

My goal is to maximize an objective function using coordinate descent over a 3-dimensional vector. In the simple case the domain over which I am maximizing is defined as follows: $X \in \mathcal{X}$ ...
-2
votes
0answers
12 views

squares a diagonal line of a rectangle passes through [closed]

a rectangle, whose length and width are whole numbers, is drawn on 1 centimeter grid paper. a diagonal line is drawn from the bottom let to the top right. how many squares does the diagonal pass ...
-1
votes
0answers
28 views

Please help me to understand the following questions [closed]

I tried several times to find answers to these questions but I do not understand anything,Please help me; 1) the weyl group has two definition, first it's defined to be the quotient N(T)/T (where T ...
1
vote
2answers
43 views

Is there a closed-form solution for tangent circle in lens of two other circles?

I am given real values $p, s, t, u$ and wish to find unknown values $r, v$. As shown in the diagram below, $p$ and $s$ are radii of two given circles, with centers at $(0,-p)$ and $(0,t)$. At ...
1
vote
2answers
36 views

Existence of the inscribed hypersphere of a simplex

Letting $\textbf{T}=[u_0,...,u_n]$ be a $n$-simplex of $\mathbb{R}^n$, how does one prove the existence of the inscribed hypersphere ? Looking at the possible duplicates, people only seem to ask what ...
0
votes
2answers
51 views

In a quadrilateral ABCD ,which is not a parallelogramm. On the rays AB,CB,CD,AD we put… [closed]

In a quadrilateral $ABCD$, which is not a parallelogram, on rays $AB$, $CB$, $CD$, $AD$ we put points $K$, $L$, $M$, $N$ such that $KL\parallel MN\parallel AC$ and $LM\parallel KN\parallel BD$. Prove ...
1
vote
0answers
23 views

Circle can always represented as $\{z:|z-c|=k|z-d|\}$

I am reading Howie's Complex Analysis. There I see this remark: The observation that $c$ and $d$ are inverse points is the key to showing that every circle can be represented as $\{z:|z-c|=k|z-d|\}$...
-1
votes
1answer
41 views

Plane distance from origin [closed]

Let the plane $V$ be defined by $ax+by+cz+d=0$ with $(a,b,c)$ not equal to $(0,0,0)$. Show that the distance between $V$ and origin is $\frac{d}{\sqrt{a^2+b^2+c^2}}$.prove the statement.
0
votes
1answer
8 views

can a 3D space be parallel/perpendicular to a plane or a line or another 3D space

As title. When I thought about parallel and perpendicular between line and line, plane and plane, line and plane, I want to go a step further find a 3D space having the relationships with another 3D ...