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
52 views

Dimension of Hyperplane

Why the dimension of a of N dimensional space hyperplane is N-1? Is there a mathematical ...
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2answers
143 views

Is it true that the formulas of the volume and surface area of an n-dimensional sphere are best expressed in terms of $\eta = \frac{\pi}{2}$?

Someone told me that the formulas of the volume and surface area of an n-dimensional sphere get simplified a lot if we express them in terms of $\eta = \frac{\pi}{2}$ instead of $\pi$. . In terms of ...
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0answers
105 views

Moving up the Y axis the length of the hypotenuse of a right triangle

If I have a right $\triangle ABC$ with $B$ being the right angle and length $AB = 50$ and length $BC = 50$. Based on the Cartesian coordinate system if I wanted to move up the Y axis the length of the ...
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1answer
91 views

Geometry reference-request

I would like some reference regarding geometry. I'm a student of civil engineering, and I wanted some insight in general geometry, e.g, know what originated it, what is valid today, etc. In specific, ...
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4answers
294 views

rank($A$)=rank($A^T$) [duplicate]

Is there an elementary explanation of why the row-rank of a matrix equals its column-rank (without using adjoint maps, resp. lots of technical computations)? What is the geometric intuition behind ...
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3answers
128 views

Proof for inequality with $a,b,c,d$ with $d =\max(a,b,c,d)$

Let $a,b,c,d$ positive real numbers with $d= \max(a,b,c,d)$. Proof that $$a(d-c)+b(d-a)+c(d-b)\leq d^2$$ I believe that the GM-AM inequality with $n=4$ variables might be helpful. ...
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1answer
40 views

How to Find the Remaining Length of a Cone With Only a Part of It

I took three measurements for a certain plastic cup in my kitchen. One was of the circle on the bottom of the cup, and the other was the top(the larger opening) and the height in between the two. ...
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1answer
93 views

Square Line Picking

The probability density function of the distance between two points chosen randomly on the unit square is given by: $ P(\ell) = \begin{cases} 2\ell\left(\ell^2 - 4\ell + \pi\right) & 0 \leq \ell ...
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4answers
273 views

What is the most fundamental trigonometric function: cosine or sine? [closed]

$$\cos(\theta) = \sin \left(\tfrac{\pi}{2} - \theta\right)$$ $$\sin(\theta) = \cos \left(\tfrac{\pi}{2} - \theta\right)$$ Both are the same entity. But is sine the copy of cosine, or is cosine the ...
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1answer
43 views

Representation of cardiod in the complex plane

I noticed that the complex function $$f(z) = \frac{2}{(z+i)^2}$$ seems to map the real line onto the cardioid given by the polar equation: $$r = 1- \cos(\theta).$$ I was wondering if there is a simple ...
5
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1answer
224 views

Proof of Ptolemy's inequality?

Can anyone prove the Ptolemy inequality, which states that for any convex quadrulateral $ABCD$, the following holds:$$\overline{AB}\cdot \overline{CD}+\overline{BC}\cdot \overline{DA} \ge ...
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1answer
62 views

Find the equations of the circles that have centre $(0,0)$ and touch the circle $x^2 + y^2 - 8x - 6y + 24 = 0$

Find the equations of the circles that have centre $(0,0)$ and touch the circle $x^2 + y^2 - 8x - 6y + 24 = 0$ So far I have said: As the circles have centre $(0,0)$ their equations are of the form ...
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1answer
58 views

Why half coversed or coversed trigonometric functions are being deprecated?

As you can see here there are some names for some trigonometric functions that I can't find in any text or math related papers today. In my opinion this kind of approach will also make it easier to ...
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1answer
72 views

Cardinality and the set of all convex polygons in $\mathbb{R}^2$.

I am asked to prove that the set $\mathcal{T}$ of all convex polygons in $\mathbb{R}^2$ has cardinality equal to $2^{\omega}$. $\textbf{My Attempt:}$ I have previously shown that the set of all open ...
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1answer
41 views

Are maps locally preserving collinearity homographies?

Question Suppose $D\subseteq\mathbb R^2$ is an open disc (or generally a simply connected domain with a good boundary), and $f\colon D\to D$ is a bijection such that the images of collinear points ...
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4answers
3k views

Area of a part of a square

This question appears in the anime series Steins;Gate. Is the value of $A$ uniquely determined by the three other areas, and the fact that the eight segments on the edges have the same length? If ...
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2answers
234 views

what is the minimum value of the angles inside these triangles?

Question: Given certain points on a square(including its sides),let these points and the verteces of the squares be the verteces of a certain number of smaller triangles, no vertices of a smaller ...
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2answers
82 views

2 calculus questions with integration - check me

I have 2 questions I would like assistance with. 1) Find the area of the region bounded by the graphs $y=5x, y=15x, y=\frac{4}{x}, y=\frac{8}{x}$ This was very difficult and tedious. I had trouble ...
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2answers
178 views

Projecting 3D Point to Plane

I have a plane defined by the equation $Ax + By + Cz + D = 0$. It does not pass through the origin. I have projected the origin of my global coordinate system onto the plane, so it is at $(a, b, c)$. ...
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1answer
200 views

Average Degree of a Random Geometric Graph

A set of $N$ points are distributed randomly on a unit square with uniform distribution. Two points $\mathbf{p}_i$ and $\mathbf{p}_j$ are said to be connected if $\|\mathbf{p}_i - \mathbf{p}_j\| \leq ...
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2answers
759 views

What is the geometric meaning of the transformation of R2/R3 when every vector is multiplied by −1? Is it a rotation?

I'd imagine a sphere with the center at the origin and all length of the vectors equals the radius. But I can't imagine what would happen if all the vectors is multiplied by -1, what would it be? I ...
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2answers
47 views

What is the value of x?

I tried solving it but I just don't get the answer, does anybody know what the answer is or how to solve it?
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4answers
120 views

How to prove U•V = |U|•|V|cos(θ), if θ is the angle between |U| and |V| [duplicate]

This is a snippet from my book. How did they get from |U|$^2$ = U • V = |U|•|V| |U|/|V| ?
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3answers
82 views

PUGS is a rectangle. If the equation of PU is $y=\frac{2}{3x} + 4$. What is the slope of SP?

I don't get the answer to this problem, can somebody please tell me what the answer is.
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2answers
40 views

Geometric proof and extension of |a|=|b|=|c|=a+b+c=1 => a=1 or b=1 or c=1

We have $a,b,c\in\mathbb{C}$ verifying $|a|=|b|=|c|=a+b+c=1$, we have to show that $a=1$ or $b=1$ or $c=1$. That can be rather easily proved using trigonometry formulas. Is there a way to prove it ...
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1answer
150 views

Question about Pasch's Postulate, line going through all three sides of a triangle

I've been reading the textbook Elementary Geometry from an Advanced Standpoint by Edwin E. Moise (3rd ed.). My problem with his wording of Pasch's Postulate, and then a subsequent problem which ...
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1answer
3k views

width to height formula for hexagon

Is there a formula to calculate the height (a) of a regular hexagon when you know it's width (b)? Is it possible to adapt this formula to a sum like : ...
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1answer
219 views

Why the geodesic curvature is invariant under isometric transformations?

As I know the geodesic curvature $$ \kappa_g = \sqrt{det~g} \begin{vmatrix} \frac{du^1}{ds} & \frac{d^2u^1}{ds^2} + \Gamma^1_{\alpha\beta} \frac{du^\alpha}{ds} \frac{du^\beta}{ds} \\ ...
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0answers
159 views

Geometry - Orthocenters and Circumcenter Lengths

Let O and H be the circumcenter and orthocenter of triangle ABC, respectively. Let a, b, and c denote the side lengths, and let R denote the circumradius. Find $OH^2$ if $ R = 7 $ and $ a^2 + b^2 + ...
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2answers
108 views

Axis of symmetry of a binary image

I want to calculate the axis of symmetry of a binary image. In other words I have an image that has a black irregular shaped object with a white background. I want to find the best approximation of ...
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1answer
152 views

We can tell that f is not a translation or glide reflection (hence, it must be a rotation). How?

Let A=(0,1) B=(0,0) C=(1,0) Suppose that f(A) = (0.4,1.8), f(B) = (1,1), and f(C) = (1.8,1.6). How do we prove that if its not a translate or glide, then its a rotation? Is it because since glide ...
2
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1answer
422 views

Problem of a circle tangent to three other circles [closed]

Two circles with centres A and B and radii 14 and 7 units respectively touch each other externally. M is the mid point of segment DE and is the centre of the circle with radius 21 units. The two ...
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0answers
18 views

Grab major changes along a line

I have a line where there are many points along it. I want to be able to find the major changes in the line and just grab those points rather than grabbing all the points the line has. Here's a quick ...
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3answers
57 views

Reflect the plane in the $x$-axis, and then in the line $y = \frac12$. Show that the resulting isometry sends $(x,y)$ to $(x,y+1)$

I have a hard time proving this without using any numbers. How do I show that the point $(x,y)$ reflected across $y=\frac12$ is $(x, 1-y)$ ?
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0answers
54 views

building a polytop from polytop and finding its volume

Let $P$ be a symmetric polytope with $M$ vertices. Suppose we subdivide this polytope into $M$ equal parts $A_i, i=1, \ldots, M$ such that each part $A_i$ correspond to one vertex, $v_i, i=1, \ldots, ...
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1answer
113 views

Can the intersection of triangles produce a parallelogram?

I came across a question asking for a list of figures that can be formed by the intersection of two arbitrary triangles. Attempts showed that we can get at least, lines, rectangles, squares, and ...
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1answer
81 views

Mahler volume of regular polygons

I am just computing some examples of Mahler volume $M(P_{n}) = \text{vol}(P_{n})\text{vol}(P_{n}^{\circ})$ of regular polygons $P_{n}$ and I'm dumbfounded by my nonsensical results, but seemingly ...
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1answer
169 views

What are some simple examples I can use to demonstrate the power of geometric algebra?

What are some simple examples I can use to demonstrate the power of geometric algebra over "everyday" vector algebra? An alternative way of thinking of this question might be: what example ...
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3answers
250 views

Contiguous edges of a cube (and other regular solids)

I had the desire to create a cube out of a single piece of string, where each edge is represented only once. Through experimentation it appears that this is impossible, and the closest you can get is ...
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2answers
108 views

How to determine the general polar equation of a circle

How can you determine that the polar equation $r = a\cos(\theta)$ is a circle?
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1answer
45 views

Symmetries of the set of points $S$

Consider that we have a set of points $S$ in the plane $\mathbb R^2$ or in the space $\mathbb R^3$ and we also consider the one to one mappings $f:S \to S$ which have the following property: They ...
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2answers
59 views

What is the difference between $z=0$ plane and $26z=0$ plane

I'm using this site to calculate a plane equation. The points are $(2,3,0)$, $(5,1,0)$ and $(6,9,0)$. The result is $26z = 0$ plane. Is there a difference between $26z =0$ and $z = 0$? Moreover, ...
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1answer
156 views

Prove the Gaussian curvature $K=0$

If two families of a geodesics on a surface intersect at a constant angle $\theta$, prove that the Gaussian curvature of the surface is zero, i.e. $K=0$. Please explain how to show the question. ...
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1answer
120 views

Do circles exist

So I was wondering about circles today and if they really do exsist. If you graph a circle in function mode, your equation looks like$$y=\sqrt{1-x^2}$$ Now for simple purposes lets take a portion of ...
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1answer
86 views

How to solve this geometry question?

Let $\triangle ABC$ be an acute-angled triangle; $L$, $M$, $N$ be the feet of perpendiculars respectively from $A$, $B$, $C$ to the opposite sides; $D$, $E$, $F$ be the midpoints of the sides $BC$, ...
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3answers
159 views

What is linear, numerically and geometrically speaking?

For as simple as it is, I never fully grasped what mathematicians and physicists mean with linear . Intuitively anything that looks like a straight line is interpreted as linear, like something in ...
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0answers
75 views

Closest Points on Two Triangles in 3D Space

I have two triangles in 3D space, defined by 3 (x, y, z) points each. I'm looking to find the closest points between the two triangles, whether that be on surface, edge, or point. I'm unsure how to ...
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1answer
79 views

What is the shape of all convex combinations of $\geq$ five vectors in $\mathbb{R}^3$?

The convex combinations of two linearly independent vectors in $\mathbb{R}^3$ span a line. The convex combinations of three linearly independent vectors in $\mathbb{R}^3$ span a solid triangle. The ...
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3answers
118 views

What *is* the working form of a 1-vector in geometric algebra?

Consider the geometric algebra definition of 0-vectors (scalar), 1-vectors (vector), and the inner product. Let $a$ and $b$ be 1-vectors. Then $a + b = c$, where $c$ is another 1-vector. Now, ...
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2answers
101 views

Parallelogram Ratios

Let ABCD be a parallelogram. Let M be the midpoint of AB and N be the midpoint of AD. Diagonal BD intersects CM and CN at P and Q, respectively. Find PQ/BD. Can this be solved using similar ...