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

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Geometry problem on ratios of areas.

The vertices of a smaller square are the trisection points of a larger square. What is the ratio of the area of the smaller square to the area of the larger square. Given that the smaller square is ...
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4 views

Triangles form a harmonic set with their medians and altitudes

In a triangle $\triangle ABC$, let $AD,BE,CF$ its altitudes and $AK,BL,CM$ their medians. Show that $D\{EF,AB\} = -1$ and $K\{LM,AB\} = -1$ I don't get any of the problems here. Not any of these ...
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0answers
4 views

Locus if orthocenter

To a circle of radius $1$, two tangents are drawn from any point $P$ on a line $3$ units away from its center. They touch the circle in $A$ and $B$. Find the locus of the orthocenter of ...
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1answer
10 views

Calculate projection of a line in a square

Said that we have two points (P1, P2) that form a line, and 3 points (S1,S2,S3) that form a square, how would we calculate the position X and Y of the point resulting from the intersection of the line ...
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1answer
27 views

How to find the length of a line segment in a rectangle

there is a rectangle abcd (vertexs) and there is point labeled P inside the rectangle. AP=55 PD=60 PC =33 what is PB
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17 views

Geometric interpretation of linear forms in the sum of four (or eight) squares identity

There is a well-known sum-of-squares identity $$(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2. \tag{1}$$ Moreover, letting $\vec{u}=[\begin{smallmatrix}a\\b\end{smallmatrix}]$, ...
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1answer
21 views

How to identifiy $V \wedge V$ with the space of all alternating bilinear forms

Let $\{ e_i \}$ be a basis for $V$, then the space of tensors $V \otimes V$ could be identified with the space of all formal sums $\sum_{ij} \alpha_{ij} (e_i, e_j)$ (I know a base independent approach ...
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1answer
16 views

Topology of metric completion of Euclidean metric

Lets consider $\cal{M}=\mathbb{R}^{2}\backslash\{(0,y)\}\text { with } \{|y|\le1\}$ with the Euclidean metric with line element $ds^{2}=dx^{2}+dy^{2}$. Now consider the distance function given by ...
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21 views

What's area of a triangle described below.

A scalene triangle has one point on each side that divides the side so the two part's that make up the side form a ratio k. What's the area of the triangle formed by connecting those points, if the ...
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0answers
12 views

How do I find the relative coordinates of a picture of a plane in 3d space.

Important info (ABCD) is a normal coordinate system perpendicular to the plane I am examining at (EAGB). Plane EAGB does not appear in a way where I can just use a ruler to measure the change in 'y' ...
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1answer
54 views

Mathematical aspects of General Relativity

I was wondering what mathematical subjects are used in the study of the theory of General Relativity (black holes ...) I assume mostly differential geometry, Riemann Geometry ... but is there also ...
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22 views

Lattice representation of the Klein bottle

I'm looking at the space $\mathbb{R^2}/G$ where $G = \mathbb{Z^2}$ acts by $(n,m)(x,y) = ((-1)^mx+m,y+n))$ and I'm trying to show that this is a smooth surface. I am having a couple of problems. To ...
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0answers
30 views

Algorithm to compute wether a stabbing line exists for a set of line segments

Let S be a set of n segments in the plane. A line L that intersects all segments of S is called a traversal or stabber of S. Give an O(n^2) algorithm to decide if a stabber for S exists Using duality. ...
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1answer
12 views

Computing overlapping circle positions, equidistant from each other.

Hello, I am a programmer and I wanted to develop an application that would have several overlapping circles in the same location, where each circle can be selectable. Is there a mathematical way of ...
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0answers
23 views

Area of a circle shall equal the area of a square [on hold]

How can I, using bolzanos theorem, discuss the equal areas of a circle and a square? How can this be shown in a graph? Would be really grateful if any could help me! :)
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1answer
15 views

Measure of angle formed by chords and two circles

The following is a question from a practice GRE Math Subject Test: In the Euclidean plane, point A is on a circle centered at point O, and O is on a circle centered at A. The circles intersect at ...
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3answers
53 views

How to prove that tg 55º<$\pi/2$

How to prove that tg 55º<$\pi/2$? I checked it on a calculator, but how to prove it though? Is it some trigonometric substitution?
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1answer
12 views

Determine if this interpretation satisfies axiom Congruent axiom 1. If so, prove it. If not, find specific

Recall the interpretation of the rational plane: points are ordered pairs $(x, y)$ with $x, y \in \mathbb{Q}$; lines are solution sets of equations $ax + by + c = 0$ with $a, b, c \in \mathbb{Q}$ and ...
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1answer
21 views

How can I prove the existence of an octagon/decagon/dodecahedron?

I've had this question in my head recently due to my math teacher giving me this problem as a bonus on a test. So I have two regular hexagons inscribed in two distinct circles, of radius n. The two ...
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1answer
14 views

How to identify the surface of a cylindrical patch?

Consider that I have a 3D facet/patch that lies on the surface of a sphere. Taking four non-collinear, non-coplanar points that lie on the facet/patch I can find the patch's underlying sphere/surface. ...
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1answer
14 views

Proof: Minkowski sum polytope implies A and B polytopes

Suppose $A$ and $B$ are convex sets and their Minkowski sum $A+B$ is a polytope. How do you prove that $A$ and $B$ are polytopes as well?
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1answer
20 views

Triangulation of 4 points why Delaunay maximizes the minimum?

I have been going through the chapter on Delaunay triangulations from the book by DeBerg (http://www.cs.uu.nl/geobook/interpolation.pdf). In lemma 9.4, he simply says that "from Thales theorem" we can ...
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0answers
10 views

Inverse of Pascal theorem

I need a simple proof of Braikenridge-Maclaurin theorem, which is also known as the inverse of Pascal theorem (for conics). Do you know any article or book that contains this theorem's proof? Thank ...
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34 views

Are there any four-dimensional shapes in the whole wide world?

I've looked up images of a 4-D (four-dimensional) shape and they looked like there are built by using regular 3-D (three-dimensional) shapes using a regular 3-D shape connected to another 3-D shape ...
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1answer
66 views

On Proving that the first euclidean axiom is wrong [on hold]

Well, The first axiom in the euclidean geometry is "A straight line segment can be drawn joining any two points". But I think that there are points that can't be joined: In the image below, We have ...
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0answers
32 views

Surface of an onion-shaped church tower

I am wondering how to calculate surface of the church tower in the picture, for painting purposes. Especially, I am interested in the two 'onion-shaped' parts. I am thinking, that it is not really ...
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2answers
24 views

Given two 2D vectors, find the intersection of lines perpendicular to them?

Assume these vectors start at the origin. Given I know the (x,y) components of vectors v1 and v2, what's the most computationally efficient way of finding v3, which points to the location of the ...
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1answer
13 views

Rectangle division into shapes and connecting adjacent shapes with non-intersecting lines

Rectangle is divided into several non-convex shapes. Adjacent shape's centroids are connected with straight lines. For example (here centroids are approximate): Could it be that some of those line ...
3
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2answers
45 views

Beautiful little geometry problem about sines

Given triangles ABC and $A_1B_1C_1$ such that $\sin A = \cos A_1, \sin B = \cos B_1, \sin C = \cos C_1$. What are the possible values for the biggest of these 6 angles? I tried some stuff like sine ...
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0answers
7 views

Survey land mass

How many acres are contained in the tract described as "beginning at the NW corner of the SW1/4 then south along the west line to the SW corner of the section then east along the south line of the ...
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0answers
12 views

What is the are of the base and what is the volume of the box? [on hold]

I'm cutting equal squares from each corner of length X from an 18x8 piece of cardboard. What if 3 inch square is cut out of each corner? What is the area of the base? What is the volume of the box?
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0answers
14 views

Inscribed and circumscribed polygons [on hold]

Given a circle, prove (with basic geometric methods: no trigonometry) that the area of any inscribed irregular polygon is strictly smaller than the area of any circumscribed polygon. Extra ...
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1answer
17 views

Drawing toral automorphisms

How do we set about drawing a toral automorphism as in figure 5.1 in the picture above. How do we know where the points highlighted in yellow are? What happens if the eigenvalue (Im guessing some ...
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1answer
12 views

A circus tent is cylindrical to a height of 3 meters and conical above it.

A circus tent is cylindrical to a height of 3 meters and conical above it. If it's diameter is 105 meter and slant height of the conical portion is 53 m, calculate the length of the canvas 5 m wide ...
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4answers
41 views

Required to calculate the area of the square problem

I am trying to calculate the area of the square $ABCD$. I have noticed that there are many similar triangles found inside of the square with the ratio of $BE:AB = 2:3$. I am struggling to get the ...
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0answers
7 views

Gluing of two geodesic space along a proper space is geodesic.

Let $X_1$ and $X_2$ geodesic metric space glued along A a proper subspace of both and then given the pseudo metric. Why is the glued space geodesic? Any hint ? For notation and details one can see ...
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1answer
12 views

Find the equations of the two circles which pass through the point $(2,0)$ and have both the $y$-axis and the line $y-1=0$ as tangents

I have this question: Find the equations of the two circles which pass through the point $(2,0)$ and have both the $y$-axis and the line $y-1=0$ as tangents. By plotting $y=1$ and $(2,0)$, it's ...
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0answers
26 views

Find a line with measure 0

A finite measure $m$ is defined on a $k$-connected set $D$, with $k>1$. You want to convert $D$ into a $(k-1)$-connected set without hurting the measure. Formally: prove that there is a set ...
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2answers
33 views

Can Cayley-Menger Determinant Be Negative?

Cayley-Menger determinant is used to calculate the area of a triangle, volume of a tetrahedron etc. Can be seen here. My question is; If given only positive numbers, can Cayley-Menger determinant ...
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1answer
19 views

How to get an unknown point from a line

I know the coordinates of the two green dots and I know that I want the red dot to be 10 units away in the x axis. What is the fastest way to get the y coordinate of the red point?
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1answer
26 views

Find the N versors more 'spaced' [on hold]

I have to deal with a concrete problem that is: Given a 3d object I want to select N directions with N integer and N>=3 for projection that would maximize the information I gain and thus my ability to ...
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1answer
18 views

In a similar triangle, if the property is SAS the the other two property automatically becomes true?

As we know that for any two triangles to be similar, they need to have any one of the property. AAA SAS SSS My question is If the triangle holds the property AAA then does it mean that SAS and ...
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2answers
36 views

Why the two angles are equal here?

It has been given that s and t are the midpoints of PR and QR respectively. My question is how can we say that the angle STR and angle PQR are equal. Is it because ST and PQ are parallel? But it is ...
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1answer
23 views

$ABCD $ is cyclic quadrilateral.The side $AB$ is extended to $E$ .In such a way that $BE =BC$

$ABCD$ is cyclic quadrilateral.The side $AB$ is extended to $E$ in such a way that $BE =BC$.If $\angle ADC =70$ degree,$\angle BAD=95$ degree then $\angle DCE$ is ... ? Note:$ABCD$ is cyclic the sum ...
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2answers
35 views

Equation of a curved line that passes through 3 points?

I have a screen wherein the upper-leftmost part is at x,y coordinate (0,0). Then I have a curved line that passes through 3 points: (132, 201), (295, 661) and (644, 1085). Now, say I want to find 7 ...
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1answer
11 views

Number of Rotations of a unit cube

Let $C $ be the unit cube $[-1,1]^3 \subseteq \mathbb R^3$.How many rotations are there in $\mathbb R^3$ which take $\mathbb C$ to itself? Please help me to visualize this.
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2answers
32 views

Locus given by a pair of scissors sliding along the ground.

I came up with this problem when dragging a pair of scissors along the ground. The question is, more mathematically: Suppose there is a point (a,0) and a point (0,b) with a fixed distance m between ...
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0answers
11 views

Discrete set of points $\xrightarrow{to}$ Curve $=$ Parameterization? [on hold]

I have to solve a problem about generating 2D vector images from a collection of 2D points that shape a 2D figure, and also I should algorithmically find a 3D NURBS/set of curves starting for a ...
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1answer
27 views

Intersections in polygons

I'm having troubles solving the following problem which is about combinatorics: let $n$ be a natural number $\ge 3$, and a convex polygon with $n$ vertices. Each vertices are supposed to connect ...
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1answer
23 views

The line of two conjugates poles make a harmonic set

Let $A,B$ be two conjugate points respect to a circle $K$ of center $O$ and radius $k$ and let $C,D$ be the intersection points of the line $AB$ and the circle $K$. (A and B are conjugates if the ...