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
32 views

is the followng function $f$ surjective?

$f$ is a function mapping $x$ axis to plane $V$ defined by if $P(x,0)$ then $f(P) = (x,x^2)$ ? I am not so sure for my following answer : to show that $f$ is surjective, for every $A(x,x^2)$ there ...
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0answers
13 views

Torus Cylinder intersection

Two surfaces in 3D .. a Torus and a Cylinder with parametrizations respectively as: $$ \{ b + a \cos(u) ) \cos(v), (b + a \cos(u) \sin(v), a \sin(u )\}, $$ $$ \{a + b \cos(p), b \sin(p) ...
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0answers
34 views

Area of an isosceles triangle

Let us consider an isosceles triangle $\triangle AOC$ if $AO=OC$ And $OB$ and $AG$ are altitude then, $$\psi=OC\sqrt{\frac{GC\cdot OB}{2}}$$ Where $\psi$ indicates area of triangle And now i have ...
2
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0answers
29 views

what exactly is arc length element $ds$ or area element $dA$

I am reading a book on complex analysis and it has something like: The spherical arc length element on the Riemann sphere ($S^2$) works out to be $ds=\frac{|dz|}{1+|z|^2},$ and the spherical area ...
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1answer
12 views

3D Shape Name Recognition

I have 1: a right angled triangle, 2: an isosceles triangle, 3: a rectangle, 4: a parallelogram, and 5 a trapezium in the 2D picture which I am rotating about the red axis next to them. They produce ...
0
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1answer
46 views

How do I prove that sine is dependent only on the angle? [on hold]

I did the following: Taking one triangle and writing the pythagorean formula to it: $a^2+b^2=c^2$ and hence: $$\frac{o}{h}=\frac{\pm \sqrt{c^2-a^2}}{\pm \sqrt{a^2+b^2}}$$ I took another ...
0
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0answers
23 views

Given triangle w/ two congruent perpendicular bisectors, is it isosceles?

Given any triangle with two congruent perpendicular bisectors (the segment that is inside the triangle), must it be isosceles? Why? Trying to construct a non-isosceles one doesn't seem to work. I ...
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2answers
36 views

Are planes in $3$-dimensions two-dimensional?

Are planes in $3$-dimensions two-dimensional? The reason I ask is because mathematically the $xy$-plane exists in $3$D space but appears to be $2$D, but how can something $2$D be in $3$D space? I ...
1
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1answer
17 views

prove that $MN \parallel BC$ in an equilateral triangle

$\Delta ABC$ is equilateral with $M$ and $N$ being interior points. if $\angle MAB=\angle MBA=40^{\circ}$ $\angle NAB=20^{\circ}$ and $\angle NBA=30^{\circ}$. Prove that $MN \parallel BC$ from ...
0
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1answer
15 views

Conditions for convex hulls

What are the conditions for there to exist a convex hull on a set $X$ of points? I know that there exists a unique convex hull for a set of $X$ points, but must no $3$ points be collinear in ...
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1answer
28 views

How could I use the centroid and size of a triangle to find the coordinates of its vertices?

I'm making a 3D graphics program in OpenGL and I'm making a function to automatically place the centroid of a triangle on the specified xy coordinates. The triangle will be of a set size. Essentially ...
2
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1answer
24 views

Local isometry between simply connected manifolds

Suppose $D:\tilde{M}\rightarrow N$ is a local diffeomorphism between two simply connected smooth manifolds $\tilde{M}$ and $\tilde{N}$. $D$ is onto. In the case of $D$ being a covering map, it follows ...
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0answers
17 views

A proof in Desargues' geometry

The question is: Prove in Desargues' geometry that if a, b, c are three lines where a is parallel to both b and c then b and c intersect at the pole of a. Desargues' Geometry has the following ...
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2answers
43 views

minimising sum of distances

I have three points $A(-3.5, 0), B(2,0), C(0,3)$. I am looking for the fourth point $D(0,d)$ such that $AD + BD + CD$ is minimal. Fermat does not work here due to $D$ lying on the y-axis. I thought ...
0
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1answer
27 views

Centroid of heart shape

A heart shape is constructed using two identical circles with radius $r$. A line is drawn from point $T$ to $C$. There, two tangents are constructed, one to each circle, through $AC$ and $CB$. (The ...
0
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1answer
20 views

Size of a 3D object with relation to reference point [on hold]

I have a simple image of a table. I placed a reference ($10 \times 10$ rectangle) on top of it. I know the size of the rectangle and I want to calculate the size of the table. If I try simple ...
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3answers
36 views

The line $x+\sqrt{3} y-10=0$ makes an angle of $150$° with the positive sense of the $x$-axis. How can this be proven?

I cant figure out how this is correct. I know that $\tan(a)=m$ of a line but I cant figure this out. Could someone show how to prove the line makes an angle of $150$° with the positive $x$-axis? I ...
0
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0answers
21 views

tensor product of a line bundle with $\bigwedge^k T^*M$

I am reading a source that says: Let $L \to M$ be complex line bundle. Define $\Omega^k(M,L)=C^\infty(M,L \otimes\bigwedge^k T^*M)$. Can someone explain thoroughly what this means? What does it mean ...
0
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1answer
31 views

Equation of a curved line from a graph

I am trying to calculate an equation to represent the graph attached to this question. It's an extract from a take-off performance graph used in aviation. The second graph shows how it is used. The ...
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4answers
30 views

Problem based on circle geometry - related to circumcircles and angles finding the angles within a circle

Let the vertex of an angle $ABC$ be located outside a circle and let the sides of the angle intersect equal chords $AD$ and $CE$ with the circle. Prove that the angle $ABC$ is equal to the half the ...
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1answer
18 views

Convert quaternions to xyz degrees

I knew quaternion for the first time a few days ago and I still don't get the way it works even when reading explanations. All I want to do is to make a subtraction between two quaternions and convert ...
0
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1answer
20 views

Maximum hyperrectangle

Is there a way to determine the coordinates of the maximum hyper-rectangle in n-D space subject to linear constraints and $x_i\ge0$ ? Example: Argument Maximum of $x_1 x_2 x_3$ Given ...
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0answers
7 views

Faces of a Bipyramid over a a Simplicial Polytope

Is there a simple way of expressing the number of faces of a bipyramid built over a polytope that is known to be simplicial, using the number of faces of the original polytope? This seems an easy ...
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1answer
41 views

Proving the volume of sphere by using tiny volumes

How can I prove the volume of sphere, by using many cones starting at the center of the sphere? It doesn't have to be cones, pyramids also work.
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0answers
27 views

Describe curves by words

I am trying to describe the general aspect of the following curves My ideas: The curves are continuous, and defined for positive values of x, and giving negative values. Have logarithmic growth ...
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1answer
23 views

Find the equation of locus of a point which is at a distance $5$ from $A$ $(4,-3)$ [on hold]

Find the equation of locus of a point which is at a distance $5$ from $A$ $(4,-3)$. Could some explain how to solve this in detailed.
1
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1answer
34 views

Two tangents BC and BD are drawn. Prove that Ob=2BC

Two tangent segments BC & BD are drawn to a circle with centre O such that $\angle$CBD=120$^{\circ}$. Prove that OB=2BC. What I've tried, BC=BD[two tangents drawn from a single point to the ...
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3answers
54 views

Prove a geometry related equation.

$\Delta ABC$ is an isosceles triangle. $\angle BAC$ is an right angle. $BC$ is its hypotenuse and $P$ is any point on $BC$. Prove that, $PB^2+PC^2=2\times PA^2$. I have tried it in many ways and ...
0
votes
1answer
21 views

ABCD is a square of side 4cm. E is a point in the interior of the square such that CED is equilateral. Then find the area of ACE in sq.centimeters.

The given answer is $4*(\sqrt{3} - 1)$ I tried all the methods but could not match the answer. Please tell if the question is wrong. Thanks in advance.
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4answers
47 views

Move a dot along a path

I have a multi-point, straight line path - to keep it simple it has three points, A B & C. A = 60,410 B = 127.5,410 C = 195,240 This is the 'template' path, I need to animate a dot moving ...
0
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2answers
18 views

Spectral theorem for unitary operators T o F

Si $T$ es unitaria y $B$ es una base de $V$ formada por vectores propios de $T$ entonces $B$ es un conjunto ortogonal. If $T$ is unitary and $B$ is a basis for $V$ consisting of eigenvectors of $T$ ...
1
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2answers
52 views

Third Ailles Rectangle

The Ailles rectangle (named after an Ontario high school teacher, D. S. Ailles) is a rectangle of size $\sqrt{3}\times\sqrt{3}+1$ with three kind of triangles, like below. We have triangle ...
1
vote
1answer
54 views

Show that the line $KD$ bisects $\angle{EKF}$

Let $P$ be an interior point of an acute-angled triangle $ABC$. The line $BP$ meets the line $AC$ at $E$, and the line $CP$ meets the line $AB$ at $F$. The lines $AP$ and $EF$ intersect each ...
2
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1answer
24 views

Distance between projections

Let $x,y,z \in \mathbb R^2$ such that $||x|| = ||y||= ||z|| = 1$. Project $z$ onto the lines defined by $x$ and $y$ as follows: \begin{equation} z_x = (z^\text{T}x) x, \ \ z_y = (z^\text{T}y) y, ...
0
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1answer
23 views

Is the interior of a simple polygon, simply-connected?

This may be trivial, but I want to be sure I understand correctly: Is it true that the interior of a simple polygon is always a simply-connected subset of the plane? I.e, is it eligible for the ...
1
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2answers
22 views

Algorithm to cover maximal number of points with one circle of given radius

we have a plane with some points on it. We know coordinate of each point apriori. We also have a circle of unit radius. I need an algorithm that determines optimal/sub-optimal position of a circle ...
0
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0answers
15 views

find all straight-lines that goes through a point $p$ in projective and affine space [on hold]

I have an exam tomorrow in projective Geometry and I am lacking of many basic skills. For instance this one: How can I find all straight lines that go through $p$ in the projective space $\Bbb ...
0
votes
1answer
21 views

How can I prove triangle ABC is congruent to triangle MaMbMc? [on hold]

Ma is a point set halfway on AB, Mb halfway on BC, Mc halfway on CA.
3
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0answers
51 views

How to calculate the area of the visible parts of a 3D PieChart?

I have created a 3D Pie Chart whose major feat (among the others) is to be rotated: I did it to demonstrate how the visual perception of data in a Pie Chart can be distorted depending on the ...
9
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0answers
55 views

A curious triangle inequality

Let $ABC$ be a triangle. Pick a point $P$ inside the triangle. How would you show that \begin{equation} |PA|+|PB|+|PC|+\min\{|PA|,|PB|,|PC|\}\leq |AB|+|BC|+|CA|. \end{equation}
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1answer
56 views

prove that the quadrilateral $ABCD$ is a square

Given $ABCD$ a quadrilateral such that $AB\parallel CD$ and $\angle ACD=45^0, \angle A=90^0, \angle D=90^0 $ Need to prove that $ABCD$ is a square. I tried to use circles but it didn't help. Any ...
2
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0answers
37 views

Proof of Bernoulli's' result on a construction to divide any triangle into four equal parts with two perpendicular lines.

I have been reading about Jacob Bernoulli and came across this particular contribution of his. Although I have tried my best to search proofs of this result I have had no success so far. Probably this ...
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2answers
54 views

Determining the position of a perpendicular line segment connecting two parallel lines that is equidistant from 2 points

I was given this problem and I can't seem to think of a solution. Here is a possibly helpful graphic: Given two parallel lines (representing the banks of a river) and two arbitrary points $A$ ...
0
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1answer
9 views

$N$ - Dimensional Solid of Revolution.

Ok, so if you take a line, or a group of lines, and rotate them $360$ degrees along their axis, you'll get a $3$-dimensional solid. Is it possible to take a $3$-dimensional figure and rotate it ...
1
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1answer
42 views

Find the length as a function of $r_1,r_2$

We are given two mutually tangent circles in the plane, with radii $r_1,r_2$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a ...
1
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0answers
30 views

Intersection of a surface and a line.

I have a surface ($H$) that passes every corner points(one coordinate gets its maximum $1$ while others $0$), such as $(1,0,...,0), (0,1,0,...,0),....,(0,...,1).$ H is characterized by ...
1
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1answer
17 views

Signed angle difference without conditions

I've got two angles in $0 \leqslant a < 360$ and I need to find the signed difference between them which should be $-180 < \Delta < 180$. Is there a way to calculate the difference with ...
3
votes
1answer
67 views

Ratios of the major axes of ellipses inscribed in a triangle

I recently came across a question that went something like this: In a triangle with vertices (0,0), (6,0), (2,4) an ellipse is inscribed such that it has the largest area. Now it is dilated, that is ...
3
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3answers
73 views

What's the equation for a rectircle? (Perfect rounded-corner rectangle without stretching on only one dim)

The equation for a rounded square seems to be: $x^4 + y^4 = 1$ You can make the radii smaller by increasing (over the even integers) the exponents in the equation. Here's a picture: Wolfram Alpha ...
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0answers
40 views

Synthetic geometry , angles.I need some ideas

Let $ABC$ be a triangle such that $m(\measuredangle ACB)>30$ and $M$ in the interior of the triangle with $m(\measuredangle BMA)=120, m(\measuredangle BCM)=30$. Let$\{ D\} = AM\cap BC$ and $P \in ...