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

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0
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2answers
250 views

Sum of Odd Triangular numbers (closed form)

I am looking for the sum of consecutive odd triangular numbers. I am trying to relate the number of $k \times k$ rhombi in an $n \times n \times n$ equilateral triangle. While I have figured out an ...
4
votes
1answer
49 views

How does this method to find the centre work?

Say we have a conic with equation $f(x,y)=c$. My teacher says that it's centre satisfies the equations : $f_x(x,y)=f_y(x,y)=0$ (If it has a centre). She didn't give any explanation. I thought this ...
0
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3answers
71 views

Find the size of two radius at once

I got my exam on Thursday, and just got a few questions left. Anyway I would aprreciate help a lot! Can anyone please help me to solve this task? You can see the picture below. The need is to finde ...
2
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1answer
569 views

Prove the similarity of isosceles triangles…

Two similar isosceles triangles are constructed outside of an parallelogram ABCD, the first being $ABB_1$ and second $CBC_1$ i.e. $|AB| = |AB_1|$ and $|CB| = |CC_1|$. Since $ABB_1$ and $CBC_1$ are ...
0
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3answers
98 views

The sum of squares of two line segments formed by a circle and coordinate axes

We have an angle of 90° so that there are 2 points A, B on each side of the angle, O is the vertex and |OA| = |OB|. On the arc AB with it's center being in O, we pick an arbitrary point P and draw ...
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1answer
169 views

Area of intersection of two general overlapping circles

My algebra is letting me down here, I can't figure out how to arrange this equation - anyone prepared to give me a hand? The area of the intersection of two circles can be defined as $$A = r^2 ...
1
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1answer
108 views

The law of cosines for a sphere

$\cos(c) = \cos(a)\cos(b) + \sin(a)\sin(b)\cos(C)$ Prove that if $a$, $b$, and $c$ is approximately $0$, then $c^2 = a^2 + b^2 - 2ab~\cos(C)$. I wasn't sure how to prove this. One thought I had was ...
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1answer
1k views

Centers of the osculating circles along an ellipse

Consider an ellipse on the plane $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. We will use the usual parametrization: $P(t)=(x(t),y(t))=(a\cos t,b\sin t)$. Then the tangent vector is $T(t)=(-a\sin t, b\cos ...
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2answers
33 views

Let $a\in\mathbb R^n$ be a fixed point. How to prove $B(a, 1/2)\cap B(g+a, 1/2)=\emptyset$ where $g\in \mathbb Z^n-\{0\}$?

Let $a\in\mathbb R^n$ be a fixed point. How to prove $B(a, 1/2)\cap B(g+a, 1/2)=\emptyset$ for some $g\in \mathbb Z^n-\{0\}$? It sounds a silly question, and obvious one, but it is a fact I need for ...
3
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1answer
298 views

What happens to the Frenet-Serret frame when $\kappa=0$?

I was considering the following question for 3D curves: Does zero curvature imply zero torsion? I think it's reasonable, because zero curvature implies the curve is a straight line, which lies in a ...
1
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1answer
737 views

Proof that cone not diffeomorphic to plane

What is the simplest way to show that the cone $\{(x,y,z)\in\mathbb R^3\,|\,z=\sqrt{x^2 + y^2}, z\geq 0\}$ is not diffeomorphic to $\mathbb R^2$? After some comments, I realize that this question The ...
5
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3answers
3k views

Greatest number of planes we can get when dividing with lines and circles

What is the greatest number of parts a plane can be divided into using $n$ infinite straight lines? What about $n$ circles? Can you generalise this into 3-dimensional space, planes and spheres? For ...
4
votes
1answer
608 views

Maximizing distance between points

I asked a similar question on SciComp, but it is a little out of the domain, so I thought I'd give it a try here as well. Give n points, I would like to place them in a periodic box (periodic such ...
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2answers
773 views

Surface area of an elliptic paraboloid

The elliptic paraboloid has height h, and two semiaxes a, b. How to find its surface area? Does it possible to use a direct formula without integrals?
3
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1answer
323 views

Where to start for studying geometry and what way should I follow after the first step?

I'm reading Gruenbaum's Tilings and Patterns: It's curious that almost all aspects of geometry relevant to the "man in the street" are ignored by our educational systems. Geometry has been almost ...
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2answers
435 views

What do you call the part of a plane “in front of or behind” a line segment?

This is a question of terminology. Suppose we have a line segment AB in a plane. The line segment forms three "zones" in the plane, where the "middle zone" is comprised of points for which some line ...
9
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3answers
293 views

Minimum ceiling height to move a closet to upright position

I brought a closet today. It has dimension $a\times b\times c$ where $c$ is the height and $a\leq b \leq c$. To assemble it, I have to lay it out on the ground, then move it to upright position. I ...
16
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1answer
322 views

Infinite Sequence of Inscribed Pentagrams - Where does it converge?

If you draw a (not necessarily regular) pentagram, there will be a pentagon-shaped hole in the middle. You can connect points to inscribe a pentagram within that hole, and then inscribe another inside ...
5
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1answer
103 views

Every closed $C^1$ curve in $\mathbb R^3 \setminus \{ 0 \}$ is the boundary of some $C^1$ 2-surface $\Sigma \subset \mathbb R^3 \setminus \{ 0 \}$

How can I prove it? This problem looks similar to Plateau's problem - but it is much more specific. I believe there exists some elementary proof. (Proving this will help me apply Stokes' theorem to ...
4
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6answers
2k views

How to draw an ellipse if a center and 3 arbitrary points on it are given?

How to draw an ellipse if a center and 3 arbitrary points on it are given?
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3answers
132 views

Diagonal of a rectangle

I need help solving this problem: The diagonal of a rectangle is $18$ cm longer than the shorter leg. If the area is $168 \ \text{cm}^2$, find the dimensions of the rectangle.
5
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1answer
158 views

Place 12 equidistant dots on a $\pi$ / 3 curve

For a computer game (which means the origin is on the top left) I need to place 12 dots equidistantly on a circle. The curve should also go through the 3 red dots shown below: Here is how I am ...
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2answers
1k views

Number of square units in area of regular hexagon with unit circle inscribed

A unit circle is inscribed in a regular hexagon. What is the number of square units in the area of the hexagon in the simplest radical form?
1
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1answer
159 views

Prove that PQ is parallel to EF.

$\delta$ is the circumscribed circle on a cyclic quadrilateral ABCD. The centre of the inscribed circle of triangle ABC is P, and that of triangle ABD is Q. Let E denote the midpoint of arc BC, and ...
1
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1answer
295 views

Hyperbolic geometry

Post Number: 45 Posted on Friday, 22 March, 2013 - 04:48 pm: I was asked the following question and i do not have any clue on these. Could anyone help me in the beginning of this? Show that there ...
0
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1answer
44 views

Find a possible path for the excavator such that the fire doesn't burn a total area greater than $13\,\mathrm{km}^2$.

A wildfire spreads at a speed of $1\,\mathrm{km/h}$ in all directions. When the fire has burnt a circle of radius $1\,\mathrm{km}$, an excavator arrives at the edge of the fire to dig a ditch to stop ...
1
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1answer
160 views

What may be the ratio of the perimeter of the trapezium to its midline?

The diagonals of a symmetric trapezium are perpendicular to each other. What may be the ratio of the perimeter of the trapezium to its midline?
3
votes
5answers
302 views

elegant proof that $\sin(x)\cdot\cos(x)=\sin(2x)/2$

I tried for a few days to prove the identity $\sin(x)\cos(x)=\frac{\sin(2x)}{2}$ and finally got the following proof. I wanted to know if someone knew a simpler or more elegant way to proof it. ...
1
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3answers
314 views

Geometry proof problem (high school)

I have an upcoming chapter test and this was one of the practice problems. Can someone guide me? Given: Isosceles $\triangle ABC$ with $AB$ congruent to $AC$; $AD$ is not a median of $\triangle ...
0
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1answer
87 views

Does a mathematical representation for orbital rotation between two concentric vortices exist?

An orbit circumscribes Vortex 1 and is inscribed by Vortex 2 such that the orbit exist as the interface between both vortices. These vortices are pure spatial rotations in the same direction. In ...
0
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1answer
71 views

second try: Find the radius in dependency of a (the side length)

I am really hopeless with this task. I have been trying nearly all day now.It´s about finding a description for the three radius in dependency of a. somethin like r=3a for example.. My friend said it ...
3
votes
3answers
112 views

inscribed angles on circle

That's basically the problem. I keep getting $\theta=90-\phi/2$. But I have a feeling its not right. What I did was draw line segments BD and AC. From there you get four triangles. I labeled the ...
3
votes
3answers
139 views

volume of “$n$-hedron”

In $\mathbb{R}^n$, why does the "$n$-hedron" $|x_1|+|x_2|+\dots+|x_n| \le 1$ have volume $\cfrac{2^n}{n!}$? I came across this fact in some of Minkowski's proofs in the field of geometry of numbers. ...
0
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1answer
97 views

Find the radius in dependency of a (the side length)

I really need help with this task, because it´s supposed to be in my exam...I added a picture of the geomtric figure. The task is to find out about radius r in dependency of a. So at the end it should ...
4
votes
2answers
202 views

How to make 3D object smooth?

I want to make the below picture into an egg with smooth surface. For the implementation in Mathematica, please, see this thread here. This thread considers mathematical methods to achieve the goal ...
2
votes
2answers
98 views

Find the line passing thought the point $p=(1,2,0)$, paralel to the plane…

Find the line passing thought the point $p=(1,2,0)$, paralel to the plane $P=\{x,y,z \mid x+2y-z=-4\}$ and crossing the line $L=\{(x,y,z):x+2y=2, y+z=4\}$ So I've tried to put the equation of plane ...
1
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4answers
105 views

Software for solving geometry problems symbolically

I've got Maple and it's excellent when it comes to solving math problems algebraically, but is there a counterpart for geometry problems? Such software would allow me to compose drawings in 2D, ...
0
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2answers
47 views

A few questions

These are sample questions that I wasn't able to solve. The length of a tangent drawn from a point $8cm$ away from the center of circle of radius $6cm$ is If perimeter of a protractor is $72cm$. ...
1
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1answer
196 views

Analytic Geometry question (high school level)

I was asked to find the focus and diretrix of the given equation: $y=x^2 -4$. This is what I have so far: Let $F = (0, -\frac{p}{2})$ be the focus, $D = (x, \frac{p}{2})$ and $P = (x,y)$ which ...
2
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0answers
86 views

Are there 3D tilings of a 3D projective hyperplane or 3-sphere?

I noticed that pentagons tile the projective plane (a spherical dodecahedron). Something they do not do on a flat euclidean plane. Is there analogous 3D tilings (honeycombs) of a 3D projective ...
2
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1answer
256 views

Geometric brainteaser

I am still studying for my exam and now I am thinking about this brain teaser. So I would really appreciate some help from you. I did found out already, that $x$ must be $46^\circ$, because of the ...
0
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2answers
81 views

Area of a rectangular triangle

We need to calculate the area of the triangle shown in figure: The text of the problem also says that: $\sin \alpha =2 \sin \beta$. What is the area of ​​the triangle?
3
votes
2answers
163 views

topic for presenting in hyperbolic geometry

For my course work, i have to give a presentation of 20-30 min presentation in hyperbolic geometry. Can any one suggest some topic(or any interesting theorem) in this area.I want to present some thing ...
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1answer
82 views

length of large and small axis of the ellipse

I need to calculate this elipse: $c^2=4x_1^2+3x_2^2-2\sqrt{2}x_1x_2$; where $c^2=1, c^2=4$ I need to calculate direction and the length of large and small axis of the ellipse. (hint: own vector and ...
2
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1answer
628 views

Finding side and angle of isosceles triangle inside two circles

I'm having a problem that I'm not sure how to solve (or if it's even possible). It's not homework, just something I'm struggling with for a project. :) Basically, there are two circles, represented ...
2
votes
1answer
529 views

A controlled trapezoid transformation with perspective projecton

I'm trying to implement a controlled trapezoid transformation in Adobe Flash's ActionScript using the built-in perspective projection facility. To give you an idea of how the effect looks like: ...
1
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1answer
122 views

We have a triangle $ABC$. On sides $AB$, $BC$, $CA$ choosen points $A_1$, $B_1$, $C_1$, different with vertices $A$, $B$, $C$.

We have a triangle $ABC$. On sides $AB$, $BC$, $CA$ choosen points $A_1$, $B_1$, $C_1$, different with vertices $A$, $B$, $C$. Then for a specified $k_a$, $k_b$, $k_c$ we have $\vec{BA_1} = ...
4
votes
2answers
357 views

Parallelogram area using determinant

Given a Parallelogram with the co-ordinates: $(a+c, b+d), (c,d), (a, b)$ and $(0, 0)$ I have to prove that the area of the Parallelogram is: $|ad-bc|$ as in the determinant of: $$\begin{bmatrix} a ...
0
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2answers
78 views

find delta, maybe by congruent triangles?

Hey there: I think this is a rather short and easy question for you. Can anyone either way please give me a hint? Would be very lovely! I found out about almost all angles in this triangle. In my ...
2
votes
1answer
147 views

Finding the incircle of a circle sector

I'm not great at mathematics so I'm sure this is trivial to most. I have been searching around however and not been able to find how to figure out the incircle of a circle sector, or, in other words, ...