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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1answer
22 views

rectangle size vs window size

I always want the large dimension of a rectangle to fill 75% of the small dimension of a window. The rectangle can be any size and the window can be any size. How do I do this?
0
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1answer
48 views

Question about area and triangle

Problem: Consider the following diagram. in $\triangle$ABC: Areas: $\triangle$AOM = a $\triangle$POC = b $\triangle$NOC = c $\triangle$BON = d. Find the area of $\triangle$MOB and ...
0
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2answers
29 views

Question about angles of rhoumbus

Problem: Consider a rhombus (Diamond) such that each of its side is the geometric mean of its diameters. I mean if length of each side is X and the diameters a and b; then $X^2$ = a.b Find the ...
0
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0answers
16 views

decimal degrees to meters conversion

I have been asked to calculate some areas (Paralleograms, Trapeziums). The areas are defined by 4 (x,y co-ordinates in decimal degrees) Do I need to convert the decimal degrees to meters before I do ...
4
votes
2answers
48 views

Choose 3 points A, B and C in a circle O

I have to get p,q and r. p = the probability of triangle ABC is an acute-angled triangle q = the probability of triangle ABC is a right-angled triangle r = the probability of triangle ABC is an ...
1
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1answer
37 views

Geometry Question about Area and surface

Problem. According to following diagram, prove (Area of (MM'N'N)) = 1/3*(Area of ABCD)). We Know that AN = NM = MB and DN' = N'M' = M'C. and quadrilateral ABCD is not and special quadrilateral. ...
1
vote
1answer
27 views

Geometry Question About Angles (Triangle) [closed]

Let $\triangle ABC$ be an isosceles triangle ($AB = AC$ and $\angle ABC = \angle ACB = 35^\circ$). We have a point $M$ inside the triangle such that $\angle MBC = 30^\circ$ and $\angle MCB = ...
1
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1answer
27 views

How to programmatically find dodecahedron's edges as couple of vertices

I'm a newbie here, and I'm not a mathematician, so I hope you could help me. Online I found that the 20 vertex of a dodecahedron can be easily expressed as: ...
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1answer
27 views

calculating coordinates along a clothoid betwen 2 curves

I need to write a program that will calculate coordinates along the designed rails of a proposed subway. The designed polyline representing the rail is composed of lines, arcs (circular curves) and ...
0
votes
1answer
29 views

Find the point on the plane xOy [closed]

Let $A(x_1; y_1)$, $B(x_2, y_2)$ and $C(x_3, y_3)$ be three points not lying on the same straight line. Find the point on the plane $xOy$ such that the sum of the distances from it to these points is ...
2
votes
3answers
91 views

4-ellipse with distance R from four foci

I'm trying to find the equation for the generalization of an ellipse called a $n$-ellipse which has a constant distance R from four foci located at $(0,0),(0,1),(1,0),(1,1)$ Edit: As an algebraic ...
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0answers
32 views

Why is the area of a spherical disc $2\pi(1-\cos r)$?

Consider the unit sphere $S^2$ and a disc centred at $u\in S^2$. Also, let $A = \{x\in \mathbb{R}^3:x\cdot u = \cos r\}$ (where $r$ is the angle between $u$ and $v$), which is the plane whose nearest ...
0
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1answer
17 views

Constructing a point $M$

Let ${\cal C}_1$ and ${\cal C}_2$ be concentric circles, with ${\cal C}_2$ in the interior of ${\cal C}_1$. From a point $A$ on ${\cal C}_1$ one draws the tangent $AB$ to ${\cal C}_2$ ($B\in {\cal ...
0
votes
1answer
22 views

Three-dimensional curve whose coordinates are rational functions

Are there three real rational functions $f,g,h$ with no poles in $[0,1]$, such that $f\geq 0,g\geq 0,f+g \geq h \geq 0$ on $[0,1]$ and the curve $\gamma(t)=(f(t),g(t),h(t)) (t\in [0,1])$ passes ...
1
vote
1answer
24 views

Finding the radius of a sphere inscribed in a right prism

We have right prism $ABCA_{1}B_{1}C_{1}$ and points $E$, $D$ such that: $A_{1}E:EB_{1}=B_{1}D:DC_{1}=1:2$ The distance between lines $AE$ and $BD$ is $\sqrt{13}$. Find the ...
1
vote
2answers
40 views

Generating Randomly distributed points inside a given triangle

Given the cartesian coordinates of three vertices of a triangle $P_1$, $P_2$, $P_3$ I know (have simulated) that I get randomly distributed points by using this protocol: $s=\text{rand}(0,1)\quad ...
0
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0answers
10 views

Chromatic number of famous lattices

What are the chromatic numbers of some well-known lattices, e.g. $E_8$ lattice, Leech lattice? (Here, of course the chromatic number means the chromatic number of the tangency graph of a lattice.)
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3answers
46 views

A simple proof that a polygon circumscribing a circle overestimates its perimeter

Looking at the picture below, it's easy to see why the perimeter of a polygon inscribed in a circle is an underestimation of the circle's perimeter. This follows from the triangle inequality: Any side ...
0
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0answers
12 views

Continuous maps to $S^n$ without antipodal pairs are homotopic [duplicate]

Let $S^n$ denote the unit sphere in the Euclidean space $\Bbb R^{n+1}$, $X$ a topological space, $f,g:X\to S^n$ are both continuous and there doesn't exist $x\in X$ such that $f(x)=-g(x)$, show ...
-1
votes
3answers
32 views

Side $BC$ of $\triangle ABC$..

Side $BC$ of $\triangle ABC$ and straight line $PQR$ are equal and parallel then prove that $\triangle AQR=\triangle PBQ$ My Attempt If we join $RC$ we get $PRCB$ is a parallelogram. Then what's ...
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0answers
27 views

Acute triangle in regular polygon

A regular 2015-gon is partitioned into triangles by a set of non-intersecting diagonals. Prove that among those triangles only one is acute-angled.
0
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0answers
59 views

Is it possible to estimate the vertical line that bisects a circle that has undergone perspective projection?

Consider the image below, where a barbell and two red plates are rendered as a perspective projection. The black lines are part of the 3D scene. The ones on the red circular surfaces bisect the ...
0
votes
1answer
31 views

Given a chord length and distance from center find length of a different chord

A chord that is of length 18 cm is 12 cm away from the center of a circle. How far is a chord of length 10 cm from the center? I know that chord of equal distance away are equidistant from the center ...
0
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0answers
18 views

(PCA) why the eigenvector points in the most “significant” direction of the data set

In Principal component analysis, why the symmetric covariance matrix's eigenvector points in the most “significant” direction of the data set?
0
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1answer
29 views

Seeing if projected lines intersect a circle

The Problem: Let $\theta$ be the angle of a line that intersects $(0,0) $, and let ($x_0,y_0)$ be the center point of a circle, where the circle is tangent to $(0,0)$. Say we 'shoot off' various ...
0
votes
1answer
11 views

Sum of the length of the perpendiculars - property of equliateral triangles

Consider an equilateral triangle ABC P is a point on AB, Q is a point on BC Suppose we draw perpendiculars from P to other sides. Let s1 be the sum of the length of these ...
1
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1answer
81 views

Show: equality of angles in ellipse between foci and external point

first please take a loot at this: Given is an ellipse with foci $F1, F2$ and an external point $P$. Through P I have constructed two tangents to the ellipse. I need to show that: $\angle F1PB1 = ...
2
votes
2answers
29 views

Slope of axes of a General Conic Section

A General Conic Section is given by the equation $ax^2 + by^2 + 2hxy +2gx +2fy + c =0 $. Let the $\theta$ be the slope of one of its axes. Prove that : $$\tan 2\theta = ...
3
votes
1answer
105 views

Is a hypersphere of non-integer dimension a fractal?

Thanks to the gamma function the formula for the surface of a unit http://mathworld.wolfram.com/Hypersphere.html $$ S(n) = \frac{2 \pi^{n/2}}{\Gamma(n/2)} $$ allows to calculate the surface of a ...
0
votes
1answer
50 views

Working out the volume of a hole in a sphere?

I'm revising for an AS-Level Maths exam and I have come across something we have never done before: My initial thought was to use the fact that volume of a cylinder is $V = \pi r^2 h$, but this ...
0
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1answer
50 views

Show that a union of triangles can be represented as a union of separated triangles.

Let a simplex $\overline{ABC}$ in the plane $\mathbb{R}^2$ be defined as a triangle with vertices $A,B,C\in\mathbb{R}^2$ i.e. the subset $\overline{ABC}:=\{P\in\mathbb{R}^2 : ...
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0answers
17 views

Symmetric Matrix for Shape Operator?

Let $R$ be a smooth surface (smoothly embedded) in $\mathbb{R}^3$. Let $M$ be the matrix for the Shape operator of $R$ with respect to the basis $\{\partial _x F, \partial_yF\}$ for the tangent space ...
1
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0answers
9 views

Metric of the doubling i.e circular billard table

Let $D^2=B_1(0) \cup_{\partial B_1(0)} B_1(0)$ denote the doubling in $\mathbb{R}^2$ i.e. the metric space which one gets after gluing two closed balls of radius $1$ with their induced metric coming ...
1
vote
0answers
20 views

Random sphere packing density

Inside a cube, we repetedly pick random points where a sphere with a diameter of 1 can be placed, until no more spheres can be placed. We then measure the percentage of space used up by the spheres. ...
0
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0answers
15 views

How to measure alignment of a set of complex numbers

Consider a vector of complex numbers $v=(z_1,z_2,\dots,z_n)$ with $||v||=v^*v=1$. Each of the components $z_i$ represents a point in the complex plane, and all these points can be represented in the ...
1
vote
1answer
32 views

Optimizing overlap between two reference frames

Let me share this little optimization problem with you: I have two orthonormal sets of vectors on $\mathbb{R}^3$, related by some Euler angles $(\alpha,\beta,\gamma)$ (corresponding to those of the ...
2
votes
2answers
33 views

Calculating cosine of dihedral angle

Let $O,A,B,C$ be points in space such that $\angle AOB=60^{\circ},\angle BOC=90^{\circ},\angle COA=120^{\circ}$ Let $\theta$ be the acute angle between the planes $AOB$ and $AOC$. Find ...
0
votes
5answers
81 views

Can we prove that plumb line is vertical to ground?

Using a plumb line to make sure a wall is vertical for instance, is as far as I know one of the most primary tools in the sense that the very-very ancient builders used it as an instrument. I was ...
0
votes
2answers
56 views

Rectangle inscribed in a circular sector of angle 60

My apologies if this has been asked before. Given a circular sector, say of radius $r$, with internal angle $60^{\circ}$, construct a rectangle inscribed in that sector so that the length of the ...
5
votes
3answers
1k views

Prove there are 3 points on the circle having same colour [closed]

All the points of a circle are randomly coloured red or blue. Prove there are 3 points on the circle having same colour, representing an isosceles triangle.
3
votes
2answers
31 views

Finding the orientation (Clockwise vs Anticlockwise) of a well-defined arc

Given an arc, with two endpoints, a known radius, and a known center, is it possible to arbitrarily choose an endpoint as the start point, and determine if the motion of drawing the arc from that ...
0
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0answers
12 views

mathematical and geometric view of simplex method

what is the mathematical concept of entering variable and leaving variable in simplex method. I mean how does the numerical value decides it(the most negative value & the minimum ratio). And how ...
2
votes
1answer
65 views

Area of Convex hull

For every point set $A \subset R^2$, prove that in general the sum of the coordinates of $\phi(T)$ is independent of a triangulation T and is associated to the area of the Convexv_Hull(A). We ...
2
votes
1answer
52 views

In a triangle $ABC$ with side $AB=AC$ and $\angle BAC=20 ^\circ $. $D $ is a point on side $AC$ and $BC = AD$. Find $ \angle DBC$

Problem : In a triangle $ABC$ with side $AB=AC$ and $\angle BAC=20 ^\circ $. $D $ is a point on side $AC$ and $BC = AD$. Find $ \angle DBC$ Solution: $AB =AC$ So $ \angle ACB = \angle ABC$ $ ...
3
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0answers
22 views

Textbook Recommentation: Discrete Differential Geometry

are there any good books that provide a good introduction to Discrete Differential Geometry to beginners? Thanks a lot.
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0answers
14 views

Checking if vector crosses the simplex

Let assume that I have a point in $x \in \mathbb{R}^n$ Also I have a non-zero vector defined by it's endpoint attached to this point. The third thing I have is a simplex of $\dim=n$, such that the ...
0
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0answers
21 views

Construct a quadrilateral

Construct a quadrilateral $PQRS$ such that $PQ=8$ cm, $QR=2$ cm, $PS=6$ cm, $\angle PQR=90^\circ$ and $\angle QRS=60^\circ$. Construct a line parallel to $PR$ that passes through $S$ to meet $QR$ ...
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votes
2answers
19 views

Triangle Geometry Questions [closed]

Please Help. Problem 1 $M$ is the midpoint of $\overline{AB}$ and $N$ is the midpoint of $\overline{AC}$, and $T$ is the intersection of $\overline{BN}$ and $\overline{CM}$, as shown. If ...
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votes
1answer
30 views

Maths question Geometry: Soccer field [closed]

Question Four: (This contributes to a KAP A; CAJ) The diagram below shows the measurements of a soccer field layout. On the actual field the white lines are 4cm wide. Assume the ...
0
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0answers
36 views

$\beta(s) = \int_0^s B(u)\,du$ is a unit speed curve

I'm teaching myself in this differential geometry book, so please have some patience with me if I ask something obvious. Let $\alpha(s)$ be a unit speed curve with domain $(-\epsilon,\epsilon)$ and ...