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

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1answer
144 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
262 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
293 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
84 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
69 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
108 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
116 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
86 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
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2answers
175 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
94 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 ...
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4answers
100 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, ...
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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
177 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
votes
0answers
72 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
votes
1answer
231 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
votes
2answers
75 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
160 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 ...
0
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1answer
79 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
votes
1answer
512 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
430 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
121 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
322 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
votes
2answers
68 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
124 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, ...
13
votes
1answer
265 views

Proving $\pi(\frac1A+\frac1B+\frac1C)\ge(\sin\frac A2+\sin\frac B2+\sin\frac C2)(\frac 1{\sin\frac A2}+\frac 1{\sin\frac B2}+\frac 1{\sin\frac C2})$

Let $\Delta ABC$, prove that $$\pi\left(\dfrac{1}{A}+\dfrac{1}{B}+\dfrac{1}{C}\right)\ge \left(\sin{\dfrac{A}{2}}+\sin{\dfrac{B}{2}}+\sin{\dfrac{C}{2}} \right) ...
4
votes
1answer
542 views

Constructing a circle that internally tangents a circle $\gamma$ and passes through two internal points.

The full details of this problem is given as follows Construct a circle $\gamma$ with center $O_\gamma$ , and place two points $A$ and $B$ inside $\gamma$. That does not lie on the edge of ...
0
votes
1answer
165 views

question about Gaussian map

I have questions. Can anyone help me to get the idea or figure out this problem. compute the Gaussian and mean curvature for torus. notice the metric for torus is X(U,V)=((a+b cos(u))cos(v),(a+b ...
2
votes
4answers
882 views

Analytic Geometry (high school): Why is the sum of the distances from any point of the ellipse to the two foci the major axis?

I don't understand where that formula came from. Could someone explain? For example any point (x,y) on the ellipse from the two foci (-c,0) and (c,0) is equal to 2a where 2a is the distance of the ...
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vote
2answers
40 views

Angular velocity question. Help

A helicopter's main rotor has blades that are $2.75$ meters long. If it rotates at $400$ rev/min, how fast is the tip of one of the blades moving in m/s?
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votes
1answer
573 views

Volume of a pyramid.

Find the volume of the pyramid with base in the plane $z=−9$ and sides formed by the three planes $y=0$ and $y−x=3$ and $2x+y+z=3$.
2
votes
2answers
146 views

inscribed triangle with circle

Question: Suppose points A and B are all on the circle C with center O. Prove that the perpendicular bisector of segment AB contains O. Here is how my proof goes. I used proof by contradiction. ...
8
votes
2answers
632 views

Connection between algebraic geometry and high school geometry.

if there is one thing that going to math competitions has taught me it is that I suck at high school olympiad level geometry. However I often find solace in the fact that not a lot of mathematicians ...
2
votes
2answers
60 views

finding the angle $\alpha$

I am studying for an exam and I really have problems solving tasks like these. The angle $\alpha$ has to be found. I thought about drawing a line from $\alpha$ to $2\alpha$ and then finding the size ...
2
votes
0answers
74 views

$n$ points on plane with sum of squares of L2 norm = 1

Let $p_1,p_2$ be two points on the 2 dimensional plane. $|p_1|$ denotes the $L^2$ norm of $p_1$ and $\delta(p_1,p_2)$ denotes the euclidean distance between $p_1$ and $p_2$. Let $f(p_1,\ldots, p_n) = ...
2
votes
1answer
61 views

angle inside a chordal quadrilateral

I am trying to solve this problem concerning this chordal quadrilateral. I'm supposed to find out $\beta$. Help is really needed since I study for an exam. $\beta$ should be in dependency of the angle ...
3
votes
2answers
130 views

Minimal length of non-contractible loops

Not self-intersecting loops on a connected closed orientable smooth surface $S$ must have a minimal length not to disconnect it, e.g. the equators of a torus. "Not to disconnect" is - on such surfaces ...
1
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1answer
118 views

References request: Introduction to K3 surface.

Are there some good books or survey papers about K3 surface? Thank you very much.
2
votes
0answers
58 views

volumes of balls under an affine transformation

Denote by $B_t(O,\rho) \subset \mathbb{R}^t$, the sphere centered at the origin with radius $\rho$, and $B_n(O,\delta) \subset \mathbb{R}^n$, the sphere centered at the origin with radius $\delta$. ...
1
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1answer
33 views

halfspaces question

How do I find the supporting halfspace inequality to epigraph of $$f(x) = \frac{x^2}{|x|+1}$$ at point $(1,0.5)$
0
votes
1answer
462 views

How to translate a slanted cylinder? ( iso-surface geometry)

A cylinder iso-surface formula is: $ x^2 + y^2 = 1 $ If you want to move the cylinder 1 higher on the Y axis it would be: $( x^2 + (y-1)^2 = 1 $ It gets a bit weird with any cylinder which ...
5
votes
3answers
259 views

How to show that all points are inside of unit circle?

There are $n$ points on the plane. Any $3$ of them are inside of a unit circle. How to show that all points are inside of unit circle? It is needed to prove that if there is a unit circle for each ...
6
votes
2answers
149 views

Concurrency of A'L, B'M, C'N.

Need some help with the following problem. Problem: In $\triangle ABC$ the midpoints of $BC$, $AC$, $AB$ are $L, M,$ and $N$ respectively, and the points on the circumcircle opposite to $A, B,$ and ...
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0answers
95 views

How to construct an orthogonal coordinate system from a smooth planar curve?

Given a planar curve $$\gamma:\mathbb R\to\mathbb R^2, t\mapsto \gamma(t) \text{, normalized to } |\gamma'(t)|\equiv1,$$ the tangential vector $$T(t) = \gamma'(t)$$ and the normal vector $$N(t) = ...
3
votes
2answers
292 views

Problems using idea of tangential quadrilaterals

I'm writing a ~60-page paper on cyclic, tangential and bicentric quadrilaterals. I need to give some problems (with solutions) where usage of those is "hidden". There are lots of problems that use ...
4
votes
0answers
296 views

Geometric intuition for Jordan normal forms (invariant subspaces, shearing, scaling, etc.)

I'm trying to visualize what a linear operator does to a vector space if that operator can be put into Jordan normal form. For concrete motivation, let's take $V = \mathbb{R}^3$, with some linear ...
6
votes
3answers
235 views

Prove or disprove inequality: $2a^2 + 2b^2 + 3c^2 \ge 16P$

Let $a,b,c$ be the lengths of the sides of a triangle with area $P$ prove or disprove inequality: $2a^2 + 2b^2 + 3c^2 \ge 16P$
4
votes
1answer
247 views

Rotation in 4 dimensions around an arbitrary plane

Rotations in 4 dimensions are performed around a fixed plane, they can be described by $SO(4)$, which is a group of orthogonal matrices with determinant equal to 1. It is easy to derive rotation ...
1
vote
1answer
1k views

Radius of an arch given length and Area of sector

the lengh of an arc of a circle is 12cm.the corresponding sector area is 108cm^2 Find the radius of the circle.
2
votes
1answer
128 views

Relation between Hadamard product and scalar product

Is there a known relation/formula for $$(A\circ B, C)$$ where $\circ$ is the Hadamard product and $(\cdot, \cdot)$ is the scalar (euclidean) product? In particular, I have a vector $y$ and a two ...
1
vote
1answer
6k views

About vector form of a line passing through 2 points.

According to my book: Equation of line passing through 2 points with position vectors $a$ and $b$ is $$r = a + K(b - a)$$ My question: If we are given 2 points how do we determine which point is ...