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

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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 ...
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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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27 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 ...
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1answer
19 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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Prove that $a^2(p-q)(p-r)+ b^2(q-r)(q-p)+ c^2(r-p)(r-q) =4(\delta)^2$ [on hold]

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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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 ...
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1answer
14 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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24 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 ...
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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, ...
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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 ...
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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 $b$, ...
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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 ...
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29 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 ...
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find a triangle on S2 all of whose angles are pi/2

I am stuck on this question. I am not sure how specific to be but I am thinking it is the triangle formed by any three great circles passing on S2 since they pass through the origin. Please offer ...
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9 views

Rotate a vector into a plane spanned by two other vectors

In an application test that I had to do for a job recently, I was asked the following question (I quote): “Given three vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$. Compute the rotation ...
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32 views

General form of a Möbius transformation sending two points to two points and a circle to another.

Suppose I am given a circle $C$ in $\Bbb C^*$ and two points $w_1,w_2$. Given another circle $C'$ and points $z_1,z_2$, what is the procedure to find a Möbius transformation that sends $C\to C'$, ...
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Uniform vs variable geometries

Euclidean, elliptic and hyperbolic geometry are all different. But they do share a common property: every part of space is "the same". There are no distinguished points that have different properties. ...
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Spherical geometry vs elliptic geometry

Wikipedia says that "spherical geometry" and "elliptic geometry" are both the geometry of the surface of a sphere. It also asserts that these two geometries are not the same — but neglects to ...
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Meaning of a Hypersurface resulting from Lagrange Multipliers

Suppose we have a function $f(x_1,\ldots,x_n)$ that we wish to maximize under the set of $n-1$ constrictions $g_i(x_1,\ldots,x_n) = c_i$ for $i \in \{1,\ldots,n-1\}$. We write the Lagrangian ...
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33 views

Where to place a bridge over the highway?

I've got a problem to solve. I had 2 different ideas which didn't actually work for all cases. On the both sides of highway there are 2 houses K and L (as in the attached picture). Line, that passes ...
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1answer
22 views

Find the radius given only a few variables

I'm writing a program that allows someone to generate a vertical road segment in 3D given a HEIGHT and an ANGLE. The road starts off flat, curves (to the ANGLE), has a brief straight segment (SEGLEN), ...
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$[f=q]$ is closed implies $f$ is continuous. [on hold]

Define $H=[f=q]$ to be the set of points such that $f(x)=q$. Let us assume that H is closed. Then the complement of H is open and nonempty (since $f$ does not vanish identically - ""i don't understand ...
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Intersection of two hyperplanes

$G$ and $H$ are hyperplanes in $\mathbb{P}_n$ with coordinates $g=(g_0, \ldots, g_n)$, $h=(h_0, \ldots, h_n)$. How can I find a symmetric matrix $A_Q$ of a quadric $Q$ with $ Q = G \cap H$, where ...
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How to find a point at a certain distance to other points on the same line

Assuming the points A(x1,y1) and B(x2,y2) and distances between AB (d1) and AC (d2) are known. How can I find the point C(xp,yp)? Actually it has a trivial solution, writing the distance equation 2 ...
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Distinguish between left and right polygon

When using land information data such as polygons presented as arrays of points, sometimes I need to know whether a polygon's points turn clockwise or counter-clockwise. Please tell me what kind of ...
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33 views

Are 1 dimensional connected closed smooth manifolds diffeomorphic to the $S^1$? [on hold]

Are 1 dimensional connected compact smooth manifolds without boundary diffeomorphic to the $S^1$ ?
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22 views

Does the perimeter of a 2-D object “count” toward its area?

I'm writing a quick Monte-Carlo simulation for a class in Matlab in order to estimate the value of pi as demonstrated in this gif: http://en.wikipedia.org/wiki/File:Pi_30K.gif However, I'm not sure ...
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1answer
23 views

Shortest distance between two given lines (Hint)

There seems to be this question that I can't seem to be able to solve. I'm hoping someone could help me figure out how to solve it. Question: Find the shortest distance between the lines ...
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2answers
59 views

What is the angle that an Archimedean conical spiral makes with the floor?

I have a spiral in the form $$r = r_0(1-{\theta\over2\pi k }) \{r \ge 0\}$$ where $r_0$ is an initial radius, and $k$ is the number of turns. (It is a spiral that decays from $r_0$ to $0$ as $\theta$ ...
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36 views

What is the meaning of “girth” of a rectangular box?

Here's an optimization problem. A parcel delivery service will deliver a package only if the length plus the girth (distance around, taken perpendicular to the length) does not exceed 112 inches. ...
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Reflections on a sphere

There is a sphere located in a point s with radius r. The Sphere is a perfect mirror. If i'm sitting in the point c, I want to cast a ray to the sphere such that I hit the point p after bouncing in ...
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15 views

How do I find the angle measurement on a triangle that has one curved side?

I have tried taking this from a circle and measuring the angles as if the width and the height are the quarter of a circle however it is not measuring correctly. I have looked at it as if it is the ...
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14 views

Finding coordinates of a point on the unit circle

let $A,B,C,D,E$ be points in clockwise order on the unit circle.set $f(P)=\alpha x+\beta y$ where $P$ is a point having coordinates $(x,y)$.If $f(A)=10,f(B)=5,f(C)=4,f(D)=10$ what is $f(E)$? i am ...
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2answers
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Midpoint of the shortest distance between 2 rays in 3D

I would like to come with an algorithm to find the midpoint of the shortest between 2 rays a+tb and c+sd, where t and s are scalars. I have a scenario which I try to depict like this. One of the ...
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1answer
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Find the area of the trapezoid ABCD

Given: AB$\parallel$DC, AB=7cm, AD=BC=5cm and the distance between AB and DC is 4cm. Find the area of trapezoid ABCD. WHAT I TRIED: Since AB$\parallel$DC and AL=BM=4cm, the figure must be a ...
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A right triangle's incenter problem by pure geometry..

$ABC$ is a right triangle such that $\angle B= 90^{\circ}$ and $BD$ is the altitude to $AC$. Given that: $I$ is the incenter of $\triangle ABC$, $I_1$ is the incenter of $\triangle ABD$ and $I_2$ ...
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On the average length of the Steiner net for $n$ randomly chosen points in the unit square

$n$ points are randomly chosen in the unit square with respect to the uniform measure. What is the average length $L$ of the associated Steiner net (tree of minimum length through each of the $n$ ...
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Convex hulls intesection

There are 2 datasets: ${(0, 0),(−0.1, 0.1), (−0.3, −0.2), (0.2, 0.1)}$ and ${(0.2, −0.1),(−1.1, −1.0),(−1.3, −1.2),(−1, −1), (1, 1),(0.9, 1.2),(1.1, 1.0)}$ I want to show that this data isn't ...
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Calculating the length of a circular arc

In the post, How do the power-series definitions of sin and cos relate to their geometrical interpretations?, I am having trouble following the logic the blogger uses in the "Calculating the length of ...
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Point order in congruent triangles

Is the point order of the triangles really relevant when it comes to congruence? Let us assume that: $\triangle ABC \cong \triangle DEF$ Which means that there is a congruence correspondence $ABC ...
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50 views

Measuring The Star

In a regular pentagon the diagonals are joined to form a star. The star occupies what percent(%) of the pentagon's area?
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33 views

How is the area of a set of points in $\Bbb R^2$ defined?

Let $S$ be a subset of $\Bbb R^2$. If no vertical slice of $S$ contains gaps, we could define the area of $S$ through the following. $$A(S) = \int_{-\infty}^\infty\left(\sup\{y\in\Bbb R\mid (x,y)\in ...
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Are two lines parallel if there are two perpendiculars to the second line between them?

I came across a question like this: In a triangle ABC, there are 2 perpendiculars to any line 'l' passing through A from B and C namely BM and CN. D is the midpoint of BC. MD and ND are joined. We ...
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Geometric Meaning of Luna's Slice Theorem

I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful? This is Luna's Slice theorem from a ...
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Geometric problem with a lvl between TST and IMO.

$N$ is the orthogonal projection of $A$ on the side $BC$, and $K$ is the orthogonal projection of $C$ on the side $AB$, and $AN$ and $CK$ intersect in point $H$. Let $k_1$ be the circle around ...
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Is the closed unit ball of the Hilbert space homeomorphic to the unit sphere

There is a answer here: http://mathoverflow.net/questions/121685/is-the-closed-unit-ball-of-the-hilbert-space-homeomorphic-to-the-unit-sphere But I don't quite understand how the ball and the sphere ...
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45 views

Intersection of a median of a triangle with another line segment

In triangle ABC, M is the midpoint of |BC| and D is the interior point of |AB|. Point E is the intersection of the sides |AM| and |CD|. Prove that if |AD| = |DE|, then |AB| = |CE|. I know that this ...
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Definitions of Platonic and Archimedean Solids using Symmetry Groups?

A Platonic Solid is defined to be a convex polyhedron where all the faces are congruent and regular, and the same number of faces meet at each vertex. An Archimedean Solid drops the requirement that ...
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55 views

Triangle, Circle Problem

What is the area $\triangle DEF$ ? I solved it using analityc geometry. I want to see if there is way to solve it using plane geometry. I did it: $x^2+y^2=400$ $(x+10)^2+y^2=100$ I found the ...
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How do you calculate how big a 20" circle will appear when it is 500 yards away?

How can I calculate how large a 20" circle will appear at various distances (e.g., 500 yards, 400 yards, 300 yards, etc.)? Thanks!