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

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2answers
528 views

Probability that centre of the square lies inside the circle joining the two points inside the square

Two points are uniformly and independently distributed (located) inside a square. A circle is drawn such that the segment joining the two points is a diameter. Find the probability that the center of ...
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0answers
73 views

the mean width of polytopes

If the mean width of the convex set $L$ ,$$b(L)=\frac{2}{\omega_{n}}\int_{S^{n-1}} {h(L,v)} \, d \mathcal{H}^{n-1}(v)$$ How one can get the mean width of polytope?? Where the general formula is ...
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1answer
85 views

Point of tangency for a circle between two vector

I'm having two vectors p and q starting at point O (origin). These vectors are known, as well as the origin point is. I know the angle α (alpha). Given a circle with arbitrary radius r, I want to be ...
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3answers
1k views

Regional Mathematics Olympiad(RMO-India) Geometry Problem

How to do this problem? I drew the figure according to the given details but, I believe some extra lines should be drawn to solve this problem.
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4answers
309 views

Math contest geometry probability

Could someone help me with this? Suppose P is an 11-sided regular polygon and S is the set of all lines that contain two distinct vertices of P. If three lines are randomly chosen from S, what is the ...
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1answer
81 views

Find $S(\theta)$ (area of triangle)

I would appreciate if somebody could help me with the following problem Q: Find $S(\theta)$ ?
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0answers
360 views

Can this be only solved by trial and error?

The following question was asked in a competitive exam Rectangular tiles each of size 70 cm by 30 cm must be laid horizontally on a rectangular floor of size 110 cm by 130 cm, such that the ...
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3answers
476 views

Function that maps straight lines into straight lines

Consider two vector spaces $V$ and $V'$ with the same dimension. Let $f: V\longrightarrow V'$ be a bijection such that it maps straight lines into straight lines; I don't know if the following ...
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1answer
180 views

Determining a conic section from points and tangent lines

Hi everyone can you please help me with this question? Is there no shorter way to do this then my approach? Is this a correct way to do it? Determine the conic section in $\mathbb{R}P^2$ that is ...
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1answer
686 views

Solving geometry Problems

I have not done a lot of problems in geometry. But, when I looked into the Olympiad questions and answers I could find that the solution to each question include drawing extra lines(drawing normals or ...
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1answer
505 views

Circles on the plane such that every line intersects at least one of them but no line intersects more that 100 of them

I have a serious problem with this problem: Is it possible to Draw circles on the plane such that every line intersects at least one of them but no line intersects more that 100 of them !? Any help ...
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1answer
2k views

Finding the maximum area of a triangle

A triangle has integer side lengths and sum of its side lengths is 7.What is the maximum possible area of this triangle? Please give me a hint on starting with this problem.
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0answers
35 views

Vertices, cusps and osculations

Reading the Vertex (curve) Wikipedia page, the section about cusps and osculation says: Although a single generic curve will not have any higher-order vertices, they will generically occur ...
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3answers
924 views

Surface area comparison of a solid cut in half

This was just a thought I had while driving this morning, no particular application. If you make a straight line cut through a solid object such that the resulting two pieces have the same volume, ...
2
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1answer
239 views

what will be yielded by gluing the faces of tetrahedron in the picture?

Two a faces glue together according to the orientation by arrows. Two b faces glue together by the same way.what will be yielded? According to the theorem 10.1.2 of Ratcliffe's Foundations of ...
4
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1answer
655 views

Ratio of Circumference to Diameter on a sphere

I was listening to an audiobook of Einstein when they started discussing spherical geometry and how Pi was no longer the ratio of a circle's circumference to its diameter, so I set out to find the ...
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0answers
52 views

Monte Carlo model of ultra loosely packed spheres

I have two questions. I am writing two monte carlo models that randomly propagate a square area and a cubic volume with mono sized hard disks and hard spheres. In the model the only criteria is that ...
0
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1answer
77 views

proof, for a differentiable function from $\mathbb{R}^n$ to $\mathbb{R}$, $\int_{Cpq}{\nabla f\cdot\mathrm{d}\boldsymbol{r}} = f(q)-f(p)$

For a differentiable function: $f:\mathbb{R}^n\rightarrow\mathbb{R}$ prove that: $$\int_{C_{\boldsymbol{pq}}}{\nabla f}\cdot\mathrm{d}\boldsymbol{r}=f(\boldsymbol{q})-f(\boldsymbol{p})$$ where $C$ ...
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3answers
81 views

Hard about Concurrence

As shown in the figure, note that P is a point of concurrency. How we can prove it geometrically ? Any hints would be appreciated.
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0answers
31 views

Travelling along a rhumb line [duplicate]

Let's say I start at a coordinate X (consisting of a longitude and latitude) on a perfectly spherical Earth and I leave this point with a certain heading Y (ground track) for a ground distance Z, then ...
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1answer
31 views

Comparison the area

Any tricks to solve quickly that which one is bigger: area of DPC or area of DQR?
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4answers
52 views

Angle guessing in geometry

Is there any quick formula to answer that which one is greater, w or v?
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4answers
640 views

Parabola and Circle problem : The parabola $y =x^2-8x+15$ cuts the x axis at P and Q. A circle is drawn …

Problem : The parabola $y=x^2-8x+15$ cuts the x axis at P and Q. A circle is drawn through P and Q so that the origin is outside it. Find the length at a tangent to the circle from O. My approach ...
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4answers
838 views

Volume of a Square-based Pyramid

I've read previous answers that state that the volume of a pyramid is $\frac{1}{3}$ (base $\times$ height). One way to visualize the volume of a square-based pyramid is to envision a cube where every ...
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7answers
1k views

An alternative proof of 30-60-90 theorem/

A 30-60-90 theorem in Geometry is well known. The theorem states that, in a 30-60-90 right triangle, the side opposite to 30 degree angle is half of the hypotenuse I have a proof that uses ...
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3answers
2k views

How to calculate the coordinates of orthocentre.!!

How to calculate the coordinates of orthocentre.!! I was surfing through the net and got this formula.. $$x-\rm coordinate= \frac{x_1\tan A+x_2\tan B+x_3\tan C}{\tan A+\tan B+\tan C}$$ ...
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6answers
1k views

How can I show that a sequence of regular polygons with $n$ sides becomes more and more like a circle as $n \to \infty$?

If we construct regular polygons with larger and larger numbers of sides, they will look more and more like circles. That is intuitively true. I hope you will help me to express and prove it ...
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2answers
848 views

Why is the locus of the centres of the circles passing through two points is the perpendicular bisector of the two points?

Why is the locus of the centres of the circles passing through two points is the perpendicular bisector of the two points?
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4answers
276 views

Inequality for each $a, b, c, d$ being each area of four faces of a tetrahedron

We know 'triangle inequality'. I'm interested in the generalization of this inequality. Here is my question. Question: How can we represent a necessary and sufficient condition for each positive ...
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1answer
59 views

Can a perpendicular slope be calculated for a slope expressed as a decimal?

If I am given the slope of a line, expressed as a decimal, is it at all possible to find the perpendicular slope, and how?
2
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2answers
89 views

How to find the parameter transformation between two stereo graphic projection

On the sphere: $x^2+y^2+z^2=1$, there are two stereo graphic projection(from North and South respectively): $$x=\frac{2u}{u^2+v^2+1},y=\frac{2v}{u^2+v^2+1},z=\frac{u^2+v^2-1}{u^2+v^2+1}$$ and ...
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1answer
560 views

Curve stitch primitive calculation

I want to calculate the intersection points of the following image: Assume that the three points of the triangle could be located anywhere. How would I do this?
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1answer
41 views

Making the Smallest Number of Mistakes Possible

I have the following problem. I have a set of $k$ labelled points, $\left\{\mathbf{x}_i, y_i\right\}_{i=1}^{k}$, where $\mathbf{x}_i\in \mathbb{R}^{2}$, and $y_i\in\left\{-1,1\right\}$. I want to ...
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1answer
253 views

compering distances between points in different dimension

I have pairs of points, each pair is in the same dimension, and I need to measure the distance between each pair: For example: ...
2
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1answer
104 views

Plane isometries $g\ , f$ , properties of fixed points and types

Given two plane isometries $g\ , f$ and $f^{'} = g\circ f \circ g^{-1}$ prove that: If $P$ is a fixed point of $f$ then $g\left(P\right)$ is a fixed point of $f^{'}$ and if $Q$ is a fixed point of ...
2
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3answers
827 views

Find the farthest points in d-dimensional space

We have $n$ points with $d$ coordinates each and we want to find two of them for which distance between them is the biggest, in Manhattan metric. The obvious algorithm has complexity $O(n^2 \cdot d)$ ...
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3answers
231 views

Locally approximating a circle by a line segment?

The book falls naturally into two parts. Part I is concerned with the general theory of fractals and their geometry. Firstly, various notions of dimension and methods for their calculation are ...
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1answer
108 views

Construction of Square Equal to Given Polygon.

Suppose that I have some polygon, I want to show that with "finite straight line cuts" I can get a square which is equal to that polygon. Is there an algebraic proof for that? Thank you.
3
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1answer
188 views

the ratio of the following two areas

Suppose you have the following triangle $ABC$: with the following properties: $|AB|=4\cdot |AA'|$, $|AC|=4\cdot |CC'|$, $|BC|=4\cdot |BB'|$. I have to find the ratio of the total area of the triangle ...
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2answers
107 views

Keeping a dynamic-dimensioned square centered in a rectangle

For some reason this problem which seemed fairly easy on the surface has given me a lot of trouble. I have a square who's side length changes based on an independent variable. The square's upper left ...
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1answer
765 views

Find angle at point on bezier curve

I have two end points and two control points. I am using these points and this link. i have found a point on bezier curve. Now i would like to find angle at this point on bezier curve. Is there any ...
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0answers
31 views

Proof: Every H-Projection is product of H-Reflections

I have some problems with the proof of the statement above (sorry for a bad translation I guess). First of all some definitions: Definition 1 (möbius-transformation): Let $A \in \mathbb{C}^{2 ...
4
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1answer
126 views

Is there a name for this particular class of topological space?

This is a simple question, but I can't figure out the name for this class of topological space. Say you start with the affine space $\mathbb{R}^n$ for finite n, and equip it with a metric. Now, say ...
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1answer
115 views

Inclusion mapping in conformal compactifications

The conformal compactification of the prototype Euclidean space $\mathbb R^n$ can be carried out by embedding $\mathbb R^n$ into $\mathbb R^{n+1,1}$ with Minkowski signature. One then considers the ...
13
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2answers
460 views

n-simplex in an intersection of n balls

Consider a $n$-simplex, $n \geq 2$ with vertices $x_i,i=1,...,n+1$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ ...
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1answer
89 views

Similarity in triangles

Sides $AB$ and $AC$ and median $AD$ of a $\triangle ABC$ are respectively proportional to sides $PQ$ and $PR$ and median $PM$ of another $\triangle PQR$. Show that $\triangle ABC$ is similar to ...
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1answer
261 views

Comparing 2 non-linear curves

I have 2 non-linear curves having (x,y)values. The x values are varying from 0 to 127 in both the curves and y values are of different magnitude for 2 curves. How can I compare these 2 non-linear ...
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1answer
84 views

parallel transportation on submanifold

$M$ is a Riemannian manifold, $N$ is a submanifold of $M$, not totally geodesic. Given two points $p,q \in N$, let $\gamma_N$, $\gamma_M$ be the geodesics connecting $p$, $q$ in $N$ and $M$, ...
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3answers
257 views

How prove this geometry $\Delta PCA \sim\Delta PBD$

let the circle $O_{1} $ and the circle $O_{2}$ the radius of is $r_{1},r_{2}$ respectively,and the circle $O_{1}$and $O_{2}$ intersection with $A$ and $B$,and the tangent to $O_{1}$ at $C$,and the ...
0
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1answer
318 views

Chord passing through concentric circles.

A chord $AB$ of one of two concentric circles at intersect each other at $C$ and $D$. We have to prove, $AC=BD$. I am not sure what this question means by 'intersect each other', but if I am ...