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
20 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
13 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
13 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
10 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
8 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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0answers
33 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
58 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
31 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
12 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 ...
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2answers
39 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
12 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
6 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
30 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
16 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
35 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
22 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
31 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
10 views

Discrete set of points $\xrightarrow{to}$ Curve $=$ Parameterization?

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 ...
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1answer
22 views

Projected Area vs. Surface Area of a 3D Set

(In what follows, I'm making up the nomenclature as I go along, so please pardon anything nonstandard.) Suppose I have a set of points $A \in \mathcal{R}^3$ which is compact, convex, and simply ...
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1answer
17 views

Ray-sphere intersection: t-value of the intersection points

You have a sphere centered at [1,2,3] with radius 3, and a ray from [10,10,10] in the direction [-1,-1,-1]. Write the implicit equation for the sphere, the parametric equation for the ray, and compute ...
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0answers
40 views

Limits, tangents and areas. Why are these statements intuitive?

I'm reading Calculus: Early Transcendentals, by Anton, Bivens and Davis (9th edition). I'm not understanding a few things. Could someone please help me? On page 68 it says: "suppose that we are ...
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1answer
34 views

Prove that $ABC$ is equilateral

Let $D,E,F$ be points on the sides BC,CA,AB respectively of a triangle $ABC$ such that $BD=CE=AF$ and $\angle BDF=\angle CED=\angle AFE$.Prove that $\triangle ABC$ is equilateral. My attempt - Using ...
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0answers
7 views

matrix of finsler metric is a positive semidefinite

Suppose $M$ be a smooth manifold, a map $F:TM \to [0,\infty]$ is called a finsler metric on $M$ if $F_{x}(y) = F(x,y)$ be a minkowski norm on $T_{x}M$ for every $x\in M$. A symmetric matrix $A\in ...
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0answers
22 views

Prove $OD$ is the angle bisector of the angle BOC

Let $ABC$ be a non-isosceles triangle and $I$ be the intersection of the three internal angle bisectors. Let $D$ be a point of BC such that $ID\perp BC$ and $O$ be a point on AD such that $IO\perp A$D ...
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1answer
44 views

Largest regular hexagon inside a square [on hold]

What is the side of the largest regular hexagon which can be drawn inside a square of side $x$?
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0answers
19 views

Existence of half-planes with respect to regular open sets of the Euclidean plane

Let $\langle\mathrm{r}\mathscr{O},\mathord{\subseteq}\rangle$ be the complete Boolean algebra of open domains (regular open sets, these that are equal to the interior of their closure: ...
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1answer
28 views

Area of square created by intersection of segments from a square vertexes to their opposite sides

There will be an square created when we draw segments from a square vertexes to their opposite sides' middle. What is the relation between smaller square's area and the side length of the bigger ...
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1answer
22 views

Equation of a specific shape's edge?

Suppose we have such a shape: It is needed to found what this shape's edges are. I mean, this shape edges are: outer arc (upper) - we know everything we would like about this arc: radius, start ...
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3answers
15 views

Cyclic quadrilaterals - finding the size of an angle

I know this might seem like a really simple question, but I really don't understand where I am going wrong. I am familiar with cyclic quadrilaterals as well as their properties, but this question ...
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0answers
14 views

Cover points with specified amount of cuboids and minimize overlap

Given a list of Points (the coords are pure integeres), I want to cover all of them with cuboids. The Problem is, I have a limited number of cuboids I can use. Of course I would like to have a ...
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1answer
17 views

Area of a triangle.

The area of a triangle $ABC$ is $144$.Denote the midpoint of $BC$ by $P$,of $AP$ by $Q$ and of $AC$ by $R$.Calculate the area of the triangle $PQR$. I draw the picture but I do not have any idea to ...
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0answers
23 views

Calculate the distance between any points in two different circles

I have two overlapping circles (C1 and C2) for which the distance between their centers is know. Inside each circle theres's random number of points (P11... P1n and P21... P2n) for which the distance ...
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1answer
23 views

Estimating the missing points of a 3D point cloud

Consider a cloud of N points (forming a smooth 3D object), in which n points are missing. Also, consider that there is no prior knowledge about the original shape of the point cloud. The only ...
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1answer
15 views

Bi-conditional statement

Can a bi-conditional be written with these 2 statements? If 3 points are collinear then they are coplanar. If 3 points are coplanar then they are collinear.
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1answer
44 views

Moving a point around a circle

we're currently working on a game which involves a character that rotates around a point. We are using a rotation matrix to rotate a given a point (x,y) around another point by first translating to ...
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1answer
34 views

Dividing a disc into equal parts

Prove that it is not possible to divide a disc into $7$ parts of equal area by means of three straight lines. Background: I saw this question asked in a way which seemed to imply the possibility ...