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

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0answers
15 views

fibration of real projective space over complex projective space

From the fibration $$U(1)=S^1\to S^{2n+1}\to \mathbb{C}P^n,$$ can we quotient the action of $S^0$ and obtain a well-defined fibration $$ S^1/S^0=\mathbb{R}P^1\cong S^1\to ...
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0answers
15 views

fibration of complex projective space over quaternion projective space

From the fibration $$Sp(1)=S^3\to S^{4n+3}\to \mathbb{H}P^n,$$ can we quotient the action of $U(1)=S^1$ and obtain a well-defined fibration $$ S^2\to \mathbb{C}P^{2n+2}\to\mathbb{H}P^n?$$ ...
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0answers
8 views

a rectangle $ABCD$, which measure $9 ft$ by $12 ft$, is folded once perpendicular to diagonal AC

A rectangle $ABCD$, which measure $9 ft$ by $12 ft$, is folded once perpendicular to diagonal AC so that the opposite vertices A and C coincide. Find the length of the fold. So I tried to fold a ...
1
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0answers
19 views

Prove the lines are concurrent (using vectors)

Problem: Let $A$, $B$, $C$, $D$, and $E$ be points on a circle. For any three points, we draw the line going through the centroid of the triangle formed by these three points that is perpendicular to ...
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0answers
6 views

Range-Based Localization of a Point using LSE

Suppose that we a set of points $P = \{p_1, p_2, \ldots, p_n\}$ in 3D. The coordinates of these points are known. In addition, we have another point, called $p$, We have Euclidean distances ...
2
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0answers
19 views

The distance from the center of a circle, which is tangent to a ellipse $x^2/a^2+y^2/b^2=1$ and two parallel tangent lines of the ellipse is $a+b$

Consider the following problem: Let $E$ be the ellipse $x^2/a^2+y^2/b^2=1$ with $a>b$. Consider two tangent lines on $E$ which are parallel, say, $r$ and $s$. Let $C$ be a circle, which is ...
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0answers
12 views

Proving that a projected line on a given plane includes given point.

As can be seen in the image, we have lines that are projected on a plane. The original lines are in the xy-plane and also in parallel with the x-axis. The lines are projected through the point ...
2
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1answer
20 views

Smallest Convex Hulls of Certain Sets in $\mathbb{R}^n$

"Given $n$, find the minimal value of $k$ with the following property: Any $k$ (distinct) points in $\mathbb{R}^n$ can be partitioned into two disjoint subsets so that the intersection of the convex ...
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0answers
20 views

Determine nodes inside a curve

Let $$x=0.5\cos(t)-0.3\cos(3t)$$ $$y=1.2+0.6\sin(t)-0.07\sin(3t)+0.2\sin(7t)$$ How could I know an arbitrary point is inside or outside of this curve? Also, another ...
4
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1answer
48 views

How can we draw a line?

We want to join by a line two distinct points $A$ and $B$. We have only a ruler of length $l>0$ and a pen. If $AB>l$ how can we do this?
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1answer
33 views

The Sine Law: A Simplified Criterion for the Ambiguous Case?

Here is my suggestion for an issue that doesn't seem to be handled well in any online notes that I have seen. Can anyone give a counter-example? If you are given $a,b,$ and $B$ in $\triangle ABC$ ...
0
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0answers
16 views

Does there exist a spherical quadrilateral with all angles pi/2?

Does there exist a spherical quadrilateral with all angles pi/2? I do not think so but I am not sure. I am unable to really visualize this. Please offer suggestions. Thank you.
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0answers
39 views

Do you have any idea about problem 4? [on hold]

Question 4: Let $\Gamma$ be a discontinuous, fixed point free group of glide reflections and translations. Let $g$ be a glide reflection of minimal length in $\Gamma$, and let $h$ be an element of ...
2
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0answers
23 views

Prove that two segments are congruent in the arbelos

Background Info + Problem I teach HS Geometry to middle school age students. I generally like to try to solve problems instead of looking up the answer, but this week a student emailed me a problem ...
2
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1answer
31 views

Definition of angle between vectors in spaces with dimensions n

I am working on a problem which requires me to find certain values of the components of vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^4$ such that the angle between them is $\pi/3$ If my ...
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0answers
23 views

Incorrect angle detected between two planes

I want to calculate the angle between 2 planes, Reference plane and Plane1. When I feed the X,Y,Z co-ordinates of pointCloud to the function plane_fit.m (by Kevin Mattheus Moerman), I get the output ...
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1answer
31 views

Using several curves in 3D to create a surface

I have a set of several closed curves in 3d (like image below is showing my set of curves from 3 views). To clarify my idea, i ask my questions in two different ways showed by diction 1 and diction ...
2
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1answer
35 views

On the Formula: $V-E+F=2$

I am looking back at the history of the formula "$V-E+F=2$" for any polyhedron in Euclidean 3-space ($V=$number of vertices; $E=$ number of edges, $F=$ number of faces). Cauchy gave following proof of ...
1
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1answer
33 views

Feyman's Triangle? How do you find the area of the inner triangle if the outside triangle is equilateral

If triangle $ABC$ is equilateral,$BD/BC=1/3, CE/CA=1/3,$ and $ AF/AB=1/3$. What is the ratio of the area of triangle? I have problems analyzing this triangle I tried to use phythagorean, heron's ...
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1answer
17 views

How to prove the bisector vector of the angle formed by three consecutive points on a helix is perpendicular to direction

I have a problem shown in the title. The problem comes from a paper named "DEFINING THE AXIS OF A HELIX"1, where it stated that "Construct vector PI from the origin to carbon alpha 2 (CA2). From ...
2
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2answers
42 views

Two squares and angle

can someone answer this please? I have two equal squares (picture). We know the lengths of three lines inside them. One of them is upside down. Find angle alpha. The picture is not perfect. Thanks
3
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0answers
21 views

Help with an Analysis problem involving isomorphisms and volumes.

I would like some help solving the following problem: Let $f: U \subseteq \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$ be a class $C^{1}$ function. Suppose that for some $a \in U$ the derivative of $f$ ...
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1answer
29 views

Proving ar($\triangle$ABC=ar($\triangle$ADC

The diagonals AC and BD of a quadrilateral ABCD intersect at O. If BO=OD, prove that area $(\triangle$ABC) = area($\triangle$ ADC) I proved area($\triangle$AOB)= area($\triangle$AOD) But, I can not ...
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1answer
22 views

Triangle problem about a point

Question: If D is a point on the side AB of ABC, find a point X on BC such that the triangles XAD and CAX are equal in area. My attempt: I don't actually know how do I solve this problem. I ...
4
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1answer
47 views

Constructing a cyclic quadrilateral of given sides.

Suppose we are given sides $a,b,c,d$. We need to construct a cyclic quadrilateral with the given sides. How can we do that? Thank you very much in advance Regards.
1
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1answer
31 views

Geometry and triangles problem

Question: If D be the mid-point of AB and if the internal bisectors of $\angle ADC$ and $\angle BDC$ meet $AC$ and $BC$ at H and I respectively. Prove that $HI \parallel AB$ My attempt: It is ...
3
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0answers
21 views

Probability density function of $x$ in the unit circle?

I'm trying to work out how to find the probability density function (PDF) for $x$ values on the unit circle - not within the unit circle but on the edge. The reason for doing so is that I'm trying to ...
-2
votes
0answers
25 views

Finding the circumradius of an isosceles triangle [on hold]

In triangle $ABC$, $$AB=AC$$ $$BC=48$$ If the inradius of $ABC$ is $12$, what is its circumradius?
0
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2answers
20 views

Problem with the formula for generating sides for a right triangle?

Everyone knows that in a right triangle, $a = \sqrt x$, $b = \dfrac{x - 1}{2}$, $c = \dfrac{x + 1}{2}$. Now, consider the triple $48, 55, 73$. Then, $\sqrt x = \sqrt{48^2}$ since the side has to be ...
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0answers
44 views

Every three of $n$ points is the vertices of an isosceles triangle. What is the max of $n$?

Suppose that we have $n\ (\ge 3)$ points in the three dimensional space and that every three of the $n$ points is the vertices of an isosceles triangle. Here, suppose that the vertices of an isosceles ...
0
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0answers
7 views

Equality of volume in hyper-sphere stacking.

(before I start I'd just like to say I am a high school student not a mathematician, so please go easy on be (haha)) in two dimensions, if one draws a square on a plane, and then draws a circle ...
1
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1answer
26 views

What's the meaning of the length of glide reflection?

Problem 4 of this sheet states: Let $\Gamma$ be a discontinuous, fixed point free group of glide reflections and translations. Let $g$ be a glide reflection of minimal length in $\Gamma$, and let ...
0
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1answer
15 views

Tangent property and rhombus

A circle is circumscribed by a parallelogram. prove by using tangent property that the parallelogram is a rhombus. I tried to prove that the adjacent sides of the parallelogram are equal but I lack ...
3
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1answer
42 views

Integration by parts (Differential Geometry)

I am studying the proof of a theorem and I am stuck. It says that by integration by parts we get that: For $g(t)$ a variation of Riemannian metrics wih $g'(0)=h,$ $$\int_{M} (-\Delta (tr h) + ...
2
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3answers
22 views

Find coordinates of point that intersects circle

I got a circle of 900 radius, knowing its center coordinates A(x1, y1) and got another point with also known coordinates B(x2, y2). I draw a line between point A and B. It intersects the circle in a ...
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3answers
86 views
+50

Find probability that random triangle covers centre of circumscribed circle

We are given the equilateral triangle A. On each edge of the triangle we pick a point: randomly (probability distribution is uniform) independently of others We construct new triangle B from ...
2
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1answer
30 views

How to find the number of right angled triangles with integer sides and inradius 2009 ..

Problem : How to find the number of right angled triangles with integer sides and inradius 2009 Please help on this as I am not getting any clue how to proceed this problem. I know that ...
2
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1answer
38 views

Locus of a point $P$ inside $\triangle ABC$

$P$ is a point inside $\triangle ABC$. $X$, $Y$ and $Z$ are feet of perpendiculars from $P$ on $BC$, $CA$ and $AB$ respectively. Find the locus of $P$ is $XY=XZ$ and $A \equiv (4,3)$, $B \equiv ...
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1answer
33 views

How to evenly space a number of points in a rectangle?

Say I have a rectangle, with variable width and height, for example lets use: width = 20 height = 30 I would like to put n ...
1
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1answer
25 views

Trapezoid problem.

Given Trapezoid ABCD with EF as median (Mid-segment)and with diagonals AC and BD with 2 points G and H as points of intersection of the median and the said diagonals ( G being the intersection of EF ...
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0answers
54 views

Prove that $a^2(p-q)(p-r)+ b^2(q-r)(q-p)+ c^2(r-p)(r-q) =4(\delta)^2$

If $p$,$q$,$r$ are the perpendiculars drawn from the vertices of a triangle ABC upon any straight line meeting the sides externally in D,E,F. where a,b,c are the sides opposite to angles A,B,C in ...
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0answers
13 views

Sylvester Gallai theorem in the complex space

An extension of Sylvester-Gallai theorem to the complex space $\mathbb{C}^d$ by Kelly states that if every line passes through at least 3 points in the given set, then these points have to be ...
1
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1answer
16 views

Find Latitude of point given Longitude and Great Circle orientation

Given the orientation of a great circle as the cartesian components of the normal vector $(a,b,c)$ to its plane, i.e. all points on the circle described in Earth-Centered Earth-Fixed (ECEF) ...
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0answers
34 views

Stuck in Preissmann's theorem

I am stuck on following the proof of Preissmann's theorem, whose statement is that Let $(M,g)$ be a closed connected Riemannian manifold of negative sectional curvature. Then every nontrivial ...
2
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1answer
20 views

How to represent two coordinate system transformations as one

I'm working on a system of relative euclidean coordinate systems. I'd like to define every coordinate system relative to a global coordinate system, which I'll refer to as [0]. Then, for example, ...
0
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0answers
21 views

Lovasz number and bipartite complement

Let $G=(V,E)$ be a graph on $n$ vertices. An ordered set of n unit vectors $U=(ui|i∈V)⊂R^N$ is called an orthonormal representation of G in $R^N$, if $u_i$ and $u_j$ are orthogonal whenever vertices i ...
0
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0answers
15 views

Positive definite functions defined on the embedding of a planar graph in the plane

By way of motivation: Let $f:[0,1]\rightarrow\mathbb{R}^2$ and $g:\mathbb{S}_1\rightarrow\mathbb{R}^2$ be continuous injections, where $\mathbb{S}_1$ is the circle ($\mathbb{R}/\mathbb{Z}$). Then, ...
2
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0answers
18 views

Double circle representation using ideal rubber bands

Let $V$ be a 3-connected planar graph, $V^*$ be its dual graph. Let $(C_i : i \in V)$ and $(D_p : p \in V^*)$ be a collection of circles so that if any edge $\{i,j\}$ borders any faces $a$ and $b$, ...
0
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2answers
14 views

Translate a rectangle but keep within bounds

In x,y space, given a coordinate P that is within bounds 0,0,bWidth,bHeight and a rectangle r where r has width rWidth and rHeight (and rWidth<=bWidth and rHeight<=bHeight), present an algorithm ...
1
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1answer
35 views

Is the intersection between two $n$-spheres an $(n-1)$-sphere?

It is true that the intersection between two $n$-sphere in $\mathbb{R}^n$ is a $(n-1)$-sphere if is not empty or a single point? I have tried to prove it but my only idea is to work with equations and ...