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

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3answers
14 views

How to find a missing radius in the surface area formula for a cone with just surface area number, and slant height?

If we know just the surface area $A$, and the slant height $h_s$ is there a way to find the radius $r$ of the base of a cone? The surface area formula for cones is: $$A = ( \pi \cdot r \cdot h_s ) ...
0
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0answers
13 views

Thomsen Blaschke condition

I am reading a paper (Paper 1: https://ideas.repec.org/p/cwl/cwldpp/76.html, that cites another paper ( Paper 2) for its proof. Paper 1, page 1, line 10 says : Consider the topological image G of a ...
3
votes
2answers
27 views

Show that at some time the hour hand of the first clock points to the tip of the hour hand of the second clock

Consider two round clocks of different sizes lying on a table: As shown on the picture, the clocks can be oriented differently but they are both set to the same time. The problem is to show that ...
1
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1answer
33 views

All lines that connect a point to a sphere directly

If I have a point P anywhere in space outside of a sphere of radius R, how do I identify all the points on the surface of that sphere that can be directly connected to P, such that the line segments ...
0
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2answers
75 views

What is $\theta$?

Suppose that $AB=BC=CD=DE=EA=BG=AG=AF=FE=DF=GC$. Then What is $\theta$?
6
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3answers
179 views

Succinct Proof: All Pentagons Are Star Shaped

Question: What is a succinct proof that all pentagons are star shaped? In case the term star shaped (or star convex) is unfamiliar or forgotten: Definition Reminder: A subset $X$ of ...
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0answers
9 views

Plucker bilinear product identity

In a certain textbook chapter dealing with Plucker product representations of lines in homogeneous coordinates, a certain bilinear form is introduced $$ ...
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0answers
7 views

Parametric form of conic in homogeneous coordinates

In my favorite computer vision text, the definition of a conic in homogeneous coordinates is given as a set of $x$ satisfying an equation $$x^{T} * C * x = 0 $$ where C is a symmetric 3x3 matrix. ...
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votes
1answer
13 views

Most efficient way to find spaces within a square on a 2D Cartesian grid? [on hold]

What's the most efficient way to find all spaces within a square with a given length centered on a given space in a 2D Cartesian grid? I get in over my head rapidly with math jargon so searching ...
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0answers
16 views

vector 3d rotation of a cube

I have a cube which is rotated by plane you can see it in an example here. What am I trying to achieve is algorithm that tells what is the top, face and side after a rotation is performed. And also ...
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2answers
35 views

Need help on geometry questions. Please help… [on hold]

In a triangle ABC, AB = AC and the ratio of angle A to angle B is 8:5. Find the angles. Two straight lines intersect such that the sum of a pair of vertically opposite angles formed is 280 degrees. ...
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0answers
32 views

Gauss Curvature Equation

One can find the following form of Gauss Curvature Equation in most introductory books on manifold, for example Jeffrey Lee's Introduction to Differential Geometry p564: $$\langle R(V,W)X,Y\rangle ...
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0answers
10 views

Some “facts” on oriented angles in the Euclidean affine space of dimension 2

Let $\mathcal{E}^2$ be an Euclidean affine space of dimension $2$ oriented by $R=(O,B=\{u_1,u_2\}$. Let $$\mathcal{r}=\{P \in \mathcal{E}_2 \text{ such that } \overrightarrow{AP} \in \langle u ...
1
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1answer
19 views

Rigorous definition of “oriented line” in an Euclidean affine space

Let $\mathcal{A}^n$ be an affine space of dimension $n$. For example, let's take $n=3$. A line $\mathcal{s}$ of $\mathcal{A}^3$ is an affine subspace of dimension $1$, that is: $\mathcal{s}=\{P ...
0
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0answers
10 views

Theoretical question. Inversive geometry on 'the now' equation.

I hope someone can help me out with this. This is an odd question but here it goes: If say, I wanted to do an inversion (using inversive geometry) to create infinity versions of 'the now'. Defining ...
2
votes
2answers
26 views

Dihedral angle Finding

I meet a wall while some problem solving. The wall is following question. There is a triangle ABC, The vertice A touch bottom plane and the distance from B, C to the bottom : BE = b, CD = c . When ...
1
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0answers
16 views

Tipi Shelter Formula

Autumn is quickly approaching in my neck of the woods and with it brings the start of camping and backpacking season. This year I want to work on my bushcraft skills with one particular project in ...
5
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0answers
43 views

Are there more ways to divide the “L”-shaped tromino into four congruent parts?

Recently my sister-in-law, who is training to become a high school mathematics teacher, asked me the following question: Consider the following polygon constructed by adjoining three squares of ...
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1answer
51 views

Prove a closed ball is a subset of another ball iff the triangle inequality is true [on hold]

For balls in $\Bbb{R}^n$ prove that: $$\bar{B}(a,r) \subset B(b,s) \iff \|a-b\| \lt s - r.$$
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4answers
27 views

How can I find the length to this geometry problem?

A person 6 feet tall is standing at the base of a lamp post that is 25 feet tall and then begins to walk away from the lamp post. When the person is 10 feet from the lamp post, what is the length of ...
4
votes
1answer
23 views

A question about angles in the Euclidean plane

It has long been known that an arbitrary angle (in the Euclidean plane) cannot be trisected using only ruler and compass, but that this can be done using a mechanical linkage. Given any positive ...
3
votes
1answer
21 views

Rolling a circle around a two dimensional curve

This is a sort of funny idea I had the other day, and although I expect to get a very technical answer I am fine with any intuitive explanation. Consider being given a function in the plane, for ...
3
votes
1answer
26 views

A strange property of continuous deformations of balls

Let $B$ be the closed unit ball in $\Bbb R^n$ and let $$F:B\times[0,\infty)\to\Bbb R^n,\quad F(x,t)=F_t(x)$$ be a continuous map such that $F_0$ is the identity. In other words, $F$ defines a ...
0
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0answers
22 views

Let $M_1 , M_2$ two manifolds of dimension $n_1, n_2$ and $M_1 \subset M_2$. Prove: $M_1$ is relatively open in $M_2$ $\iff n_1 = n_2 $.

Let $M_1 , M_2$ two manifolds of dimension $n_1, n_2$ and $M_1 \subset M_2$. Prove: $M_1$ is relatively open in $M_2$ $\iff n_1 = n_2 $. I have no idea where to start with this one, any help would be ...
4
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2answers
29 views

What is the graph that corresponds to $Q'_8$ generalized quadrangle ? Could you please explain this in plain english?

In this paper a table about large graphs with given degree and diameter graphs is shown: I would like to know what the adjacency list of the graph denoted by: in the table above is. Could ...
2
votes
2answers
29 views

Lines and planes - general concepts

I've come across a book that has this general questions about lines and planes. I can't agree with some of the answers it presents, for the reasons that I'll state below: True or False: Three ...
3
votes
1answer
52 views

Proving co-ordinates of an equilateral triangle are integers in a plane

Consider the 2D plane $P$ in $\Bbb{R}^3$ defined by $$P=\{x \in \Bbb{R}^3 \mid x_1+x_2+x_3=0\}.$$ Let $a$, $b$, $c$ be the vertices of an arbitrary equilateral triangle in $P$ such that all the ...
3
votes
0answers
44 views

Trisectible Angle

How do we prove that a triangle with sides $(one, x, y)$, where $x$ is any constructible length from one to three at the elliptic curve $$y^2 = x^3 -x^2 -x +1$$then the triangle possess at least ...
1
vote
2answers
33 views

If in a triangle $ABC$,$1=2\cos A\cos B\cos C+\cos A\cos B+\cos B\cos C+\cos C\cos A$,then prove that triangle will be equilateral triangle

If in a triangle $ABC$ we have $$1=2\cos A\cos B\cos C+\cos A\cos B+\cos B\cos C+\cos C\cos A\ ,$$ then the triangle will be equilateral triangle. I tried but except few steps,could not prove it. ...
2
votes
2answers
24 views

volume of a $n$-d parallelepiped with sides given by the row vectors of a matrix $A$ is the product of the singular values of this matrix $A$

Why the volume of a $n$-dimensional parallelepiped with sides given by the row vectors of a matrix $A$ can be seen as the product of the singular values of this matrix $A$? I only know in ...
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votes
1answer
58 views

Prove that $\frac{r_1}{r-r_1}+\frac{r_2}{r-r_2}+\frac{r_3}{r-r_3}=\frac{r_1r_2r_3}{(r-r_1)(r-r_2)(r-r_3)}$ [on hold]

Let $D,E,F$ be the feet of the perpendiculars from the incenter $I$ to the sides $BC,CA$ and $AB$ respectively. If $r,r_1,r_2$ and $r_3$ are the inradius of the triangle $ABC$ and radii of the circles ...
0
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0answers
33 views

Show that $a\sin 2\alpha+b\sin 2\beta+c\sin 2\gamma=0$

If the internal bisectors of the angles of the triangle ABC make angles $\alpha,\beta,\gamma$ with sides $a,b,c$ respectively then show that $a\sin 2\alpha+b\sin 2\beta+c\sin 2\gamma=0$ I tried to ...
0
votes
1answer
15 views

Given the end-vertices of two line segments, how do you calculate the point at which they intersect?

Given only the vertices of each line segment, and it's assumed they intersect, how do I calculate the point at which they intersect (in two and additionally three dimensions)?
2
votes
1answer
27 views

Euclidean Geometry (Potential Menelaus Theorem)

I have a strong suspicion that this problem applies Menelaus's theorem, but I can't see it. I also tried algebraic manipulation (such as trying to re-write BD/DC in terms of AB or CP), but to no ...
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votes
2answers
45 views

proof of isosceles triangle?

How do you prove this isosceles triangle? Given line AC is congruent to line BC Prove: Angle A=Angle B I've gotten to the angle bisector and SAS(side- angle- side), and I believe there is one more ...
1
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2answers
26 views

Smallest convex polyhedron containing integer points of a cylinder

A cylinder has height $6$ and radius $3$. The centers of the two bases are $(0,0,0)$ and $(0,0,6)$. Find the volume of the smallest convex polyhedron that encloses every lattice point inside the ...
1
vote
2answers
43 views

Show that $\csc^n\frac{A}{2}+\csc^n\frac{B}{2}+\csc^n\frac{C}{2}$ has the minimum value $3.2^n$

Show that in a $\Delta ABC$, $\sin\frac{A}{2}\leq\frac{a}{b+c}$ Hence or otherwise show that $\csc^n\frac{A}{2}+\csc^n\frac{B}{2}+\csc^n\frac{C}{2}$ has the minimum value $3.2^n$ for all $n\geq1$. ...
2
votes
2answers
37 views

If the circumcenter of the triangle $ABC$ is on the incircle of the triangle,then prove that $\cos A+\cos B+\cos C=\sqrt2$

If the circumcenter of the triangle $ABC$ is on the incircle of the triangle,then prove that $\cos A+\cos B+\cos C=\sqrt2$ How should i attempt this question?I thought over it hard but could not ...
0
votes
0answers
25 views

What is the name of a cilinder-like object based on an ellipse instead of a circle?

If I project a circle over the Z-axis, I'll get a cilinder. If I project a square over the Z-axis, I'll get a parallelepiped. If I project an ellipse over the Z-axis, I'll get a... whatsitsname? I ...
2
votes
0answers
31 views

Geometry/Trigonometry Determine angle in a Triangle [duplicate]

Triangle ABC is isosceles with BC as base, AB=AC and Angle A=20 degrees. Points D and E lie on sides AB and AC respectively, such that D lies between A and B, E lies between A and C, angle BCD=50 ...
2
votes
1answer
23 views

Jacobi determinant for high-dimensional sphere inversion

I need to find the Jacobi determinant for the unit sphere inversion in $\mathbb{R}^n$, i.e. the map given by $f(x) = \frac x {|x|^2}$ for $x\in \mathbb{R}^n$. The main problem is to figure out the ...
1
vote
1answer
17 views

determine cube orientation given one side in a perspective projection

Suppose that we are given an arbitrary quadrilateral T that does not have any parallel edges. I want to draw a cube in a three-point perspective projection such that T is one of its sides. The ...
3
votes
4answers
46 views

$R$ is the midpoint of $MN$ and the points where $AC$ intersects $MD$ and $ND$ are $P$ and $Q$, respectively. Show that $PR=QR$.

In square $ABCD$, $M$ and $N$ are points on $AB$ and $BC$, respectively such that $\angle MDN=45°$. $R$ is the midpoint of $MN$ and the points where $AC$ intersects $MD$ and $ND$ are $P$ and $Q$, ...
3
votes
0answers
56 views

Very special geometric shape (No name yet?)

I suppose this geometric shape is something very 'special'. I cannot clarify in short about being 'special', but I think this shape stands together with such special shapes like square and hexagon. ...
0
votes
0answers
28 views

Trignometry, bearings yacht race question [on hold]

In a yacht race each yacht has to sail around a set of 4 buoys, and then return to the start line in order to finish. We will assume that the buoys are just points and the start line is also a ...
0
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0answers
17 views

book suggestions on homogeneuos coordinate and projective geometry [on hold]

Will any one introduce some good books about projective geometry and homogeneuos coordinates to me?
2
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1answer
24 views

About the interior of a polyhedron

Let us consider a polyhedron in $\mathbb{R}^n$ (in this context it must NOT be bounded) $\mathcal{P} = \{ x: A \cdot x \leq c\}$ for some matrix $A$. Let $\mathcal{I} = \{ x: A \cdot x < c\}$ be a ...
-1
votes
0answers
23 views

Straight lines and its applicaton [on hold]

What are the number of lines passing through (1,1) and intersecting a segment of length 2 unit between the lines x+y=1 and x+y=3 ?
0
votes
2answers
43 views

Equilateral Triangle Property

If the vertices of a triangle have integral coordinates how to prove that the triangle cannot be equilateral ?
0
votes
1answer
23 views

How to describe the transformation that changes French flag to Russian flag?

http://www.wolframalpha.com/input/?t=crmtb01&f=ob&i=Russia%2C%20France%20flags I presume it can be described two group operators, but I'm not sure how to come up with the formal description. ...