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

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2
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3answers
69 views

$SL(V)$ and $PSL(V)$ act $k$-transitively on the space of all $1$-dimensional subspaces.

A group $G$ acts $k$-transitive on some set $X$ if for every two $k$-tupels $(x_1, \ldots, x_k)$ and $(y_1, \ldots, y_k)$ there exists some $g \in G$ such that $$ g\cdot x_1 = y_1, \ldots, g\cdot ...
0
votes
1answer
161 views

How to calculate the 'rotation' between 2 coordinates?

The title may have been slightly misleading, however I have got no idea how better to describe it. I have got 2 coordinates. Opens up paint.net If you look at that diagram, there is a line at the ...
0
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1answer
43 views

How the two length are equal?

I have understood that EDCF is a rhombus because of the conditions given. But I have got a solution for this problem they said that joining D and F , DF is equal to BC. How can this be true?
0
votes
1answer
94 views

Similar Triangles Problem For Geometry I

In quadrilateral QRTS, we have QR = 11, QS = 9, and ST=2. Sides RQ and ST are extended past Q and S, respectively, to meet at point P. If PS = 8 and PQ = 5, then what is RT?
1
vote
1answer
54 views

Finite family of subtori in the torus $(S^{1})^{n}$

Working on a problem on matroids, I've already ask a question about some subtori. Here's the link to a previous problem: Topological subspace in $(S^{1})^{n}$ Anyway, here's another problem related ...
1
vote
2answers
115 views

Locus of vertex of triangle moving inside circle

A right triangle with sides $3,4$ and $5$ lies inside the circle $2x^2+2y^2=25$. The triangle is moved inside the circle in such a way that its hypotenuse always forms a chord of the circle. The locus ...
2
votes
3answers
51 views

$\mathbb{R}^2$ rotations

Considering $\mathbb{R}^2$ for the two rotations $g_1,g_2$ with centers of rotation $x_1$ and $x_2$ by $\theta_1$ and $\theta_2$ I have to show that $g_1\circ g_2$ is a rotation iff $\theta_1+\theta_2\...
0
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1answer
148 views

equilateral triangle area geometry

The triangle on the left is equilateral. We produce the figure on the right by following a simple rule. We divide each side into three segments of equal length and add an equilateral triangle to the ...
0
votes
1answer
63 views

get points out of two lines where y distance is given value

there are two lines form 3 point like so Point 1 (1,2) Point 2 (5,6) Point 3 (7,1) Lines are from 1 to 2 and from 1 to 3 what i need are the points on each line where the ...
1
vote
1answer
139 views

Drawing a nested epicycloid

I would like to learn how to draw this kind of pictures (possibly with Mathematica, as it is the only language I would be comfortable to code such a thing in): There is something similar on the ...
2
votes
2answers
116 views

how to find iso-cost contours on a 2d plot efficiently

Consider a 2D plot in which dimension 1 and 2 represent quantity 1 and 2 respectively ranging over 0 to 100. Each point in the space corresponding to (x,y) represent cost of choosing quantity 1 as x ...
4
votes
4answers
996 views

True or False: The circumradius of a triangle is twice its inradius if and only if the triangle is equilateral.

Let $R$ be the circumradius and $r$ be the inradius. The if part is clear to me. For an equilateral triangle, the circumcentre, the incentre and the centroid are the same point. So, by property of ...
0
votes
3answers
53 views

Geometry problem, median,altitudes

I tried to use the angle property by which AD=4 and DB=5,but since F is not given as mid point I don't know how to proceed to find length of DG.I think AED as 90 degree is important but I am unable to ...
1
vote
2answers
203 views

Equation of the plane that passes through a line and it's parallel with another line

What is the equation for plane $P$ that passes through $D_1:( x = 1+2t, ~y=t,~ z=t )$ and its parallel with $D_2 : ( x = -t,~ y=t, ~z = 2+3t )$? I'm just learning this. I think that if $P$ plane ...
4
votes
1answer
230 views

Algebraic proof of Ehrhart's theorem

Let $P \subset \mathbb{R}^d$ be a $d$-dimensional polytope, where all vertices lie on integral coordinates, and let $L(P,n)$ denote the number of integral lattice points contained in the scaled ...
2
votes
2answers
226 views

Trying to understand formula for counting regions of hyperplane arrangements in $\mathbb{R}^2$

There are up to $\binom{n}{2} + n + 1$ regions created by a hyperplane arrangement in $\mathbb{R}^2$ containing $n$ hyperplanes. I want to understand this in a demonstrative way. Each hyperplane ...
0
votes
1answer
79 views

Proving Sin Cos Tan

I am asked to prove the following: $$\dfrac{1-\cos x}{\sin x}=\dfrac{\sin x}{1+\cos x}=\tan\dfrac x2.$$ Looking at the answer I am not able to see what is going on here: $$\frac{1 - \cos(x)}{\sin(...
2
votes
2answers
185 views

Why is the max. number of intersections of k lines in $\mathbb{R}^2$ = $\binom{k}{2}$?

Why is the maximum number of intersections of k lines in $\mathbb{R}^2$ = $\binom{k}{2}$?
0
votes
1answer
120 views

Maximize area of a rectangle between parabola and a line

I was given a task to maximize the area of a rectangle that can be inscribed between parabola $y=1-x^2$ and a line $y=0$ such that one side of the rectangle lies on the $x$ axis. My idea is to somehow ...
4
votes
1answer
266 views

A Modern Alternative to Euclidean Geometry

First of all, I want to master Geometry, I have knowledge on high school geometry and I was thinking of learning Euclidean Geometry. I bought a copy of Euclid's Elements, it is very interesting, ...
0
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3answers
40 views

Finding Height and Base of a Triangle

This is a question I got off of one of my previous math tests, and I don't even know where to start with solving it. The height of a triangle is 4 times its base. If the area of the triangle is 160 ...
1
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1answer
42 views

Order Types, Point-line duality

I am trying to understand Order Types and their enumeration. I'm having a real hard time understanding these slides.. Especially the one I am showing. Could anyone explain to me what this slide from ...
0
votes
1answer
44 views

Bounding box of a thick line with end caps

I have been pulling my hair out on the trigonometry on this and just can't seem to get it right. Basically, I need to calculate the bounding box of a line going from point (x1,y1) to (x2,y2) where ...
2
votes
1answer
63 views

weighted integral in convex hull

Working on an integral $$ J=\frac1{2\pi} \int_0^{2\pi} w(t) g(e^{it}) dt $$ where $\frac1{2\pi} \int_0^{2\pi} w(t) dt=1$ ; $w(t)$ is non-negative continuous ...
1
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0answers
790 views

Point of division of line segments one is 3/4 as the other

The segment joining A(2,-4) and B(9,3) is divided into segments one which is three fourths as long as the other. Find the point of division nearer to B(9,3). I'll call the point of division as C(x,y)....
2
votes
1answer
883 views

The locus of centre of circle tangent to two given circles

What is the locus of the centre of circles that are tangent to two given circles? I had no idea how to approach the problem so I considered a special case, namely one in which the two circles were ...
1
vote
2answers
89 views

Find length of $CD$ where $\angle BCA=120^\circ$ and $CD$ is the bisector of $\angle BCA$ meeting $AB$ at $D$

$ABC$ is a triangle with $BC=a,CA=b$ and $\angle BCA=120^\circ$. $CD$ is the bisector of $\angle BCA$ meeting $AB$ at $D$. Then the length of $CD$ is ____ ? A)$\frac{a+b}{4}$ B)$\frac{ab}{a+b}$ C)$\...
0
votes
1answer
134 views

Get a third point (lat, lng) from two given

I have two points as follow (the distance between them is variable): I need to get a third as shown: The two first points change all the time, including the distance between them. My problem: I ...
0
votes
1answer
689 views

finding points with maximum distance between them on a circle

I'm a computer science student working on a problem in computer graphics and looking for a formula that can find the x and y positions of a set of N points on the surface of a circle so that the ...
0
votes
3answers
155 views

Please find the radius of the circle.

Hello, I want to find the radius of red circle. I tried it with several ways like trigonometric. But there is a special value is given with this. I can not understand why it is given. It is the blue ...
0
votes
1answer
52 views

How to prove these conditions right or wrong?

To solve the questions, I can write easily that The area of ADC and BDC triangles equal and similarly EBD and CED triangle areas are equal. From this conditions how to show the following ...
1
vote
1answer
245 views

Finding the area of a triangle in terms of the radius of the excircle

Prove that the area of a triangle $ABC$ is $$\frac12 (b + c - a)r$$ where $r$ radius of the excircle opposite to $A$ and the rest of the symbols have their usual meaning. I started off with the ...
0
votes
2answers
367 views

solve this numerical with mass point geometry

$AD : DC = 1 : 2, BE : ED = 1 : 2$ and $EF = FC$. Find the ratio of the area of triangle $EFG$ to that of $ABC$.
1
vote
1answer
38 views

Transformation shifts parallelogram to trapezoid - fairly simple

We are given the region $D= {\{(x,y) | 1 \leq x-y \leq 2, x \leq 0, y\leq 0\} \subseteq \mathbb R^2}$ I drew this region on a piece of paper, it resembles an infinite parallelogram on the third ...
4
votes
1answer
290 views

Question about Qing Liu's Algebraic Geometry book

I was just wondering what the real prerequisites are for reading Qing Liu's 'Algebraic Geometry and Arithmetic Curves', and if it is a good first book on the subject. In his preface he states that the ...
2
votes
2answers
501 views

Forming a simple polygon from the extrusion of a polygonal chain

Let's say I have a set of vertices connected by edges to form a polygonal chain. Each vertex may be shared by a number of edges to form various sub-chains. An example is shown below. Each edge has ...
2
votes
1answer
147 views

What is a circle's area if its radius is $\pi$?

The area of a circle equals $\pi r^2$. If a circle's radius is $\pi$, what is its area? I believe the answer is $\pi^3$, right?
1
vote
3answers
79 views

How to show $n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$ has no nonzero integer solutions?

How do we prove that $$n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$$ has no nonzero integer solutions? I know two ways to prove this by taking a geometric interpretation but I don't want such a version. How ...
2
votes
2answers
198 views

A question about 4 concyclic points

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles $...
2
votes
2answers
83 views

Why are these lines tangent?

I was trying the problems at http://euclidthegame.org and for level 20, ending up using, but couldn't see the reason behind the following: We have a circle centred on B and a point A outside the ...
1
vote
2answers
45 views

A function similar to a rotated-sin

I would like to find a mathematical function like the one I sketched below. My first idea was to rotate a sin function, but now I don't think that would work because I would like the function to be ...
0
votes
2answers
104 views

Can anyone calculate the area of the top shape in this diagram?

http://postimg.org/image/w5f5moq7z/ The top shape on the diagram is the sensor that I need to calculate the area for. I have tried using the 2 elipses at the side and combining them together to get a ...
0
votes
1answer
65 views

calculate the area of this shape

It's a rectangle with 2 half elipses joined on the left and right side. The rectangle itself is 3.55 X 2.54 The width of the whole shape (rectangle with 2 elipses) is 4.195. Take away the width of ...
2
votes
1answer
61 views

total possible triangles with integral values

How many triangles with altitudes 6,8,X(unknown value) can be formed such that the value of x is an integer?
2
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0answers
118 views

Undergrad Geometry Book

What is a good follow-up to Stillwell's Four Pillars of Geometry? Also Algebraic Geometry/ Topology sounds fun -- are there any good undergrad books on that?
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votes
0answers
60 views

What does the rotation group of $\mathbb{\bar{Q}}^n$ look like?

There's a structural difference between the rotation groups of $\mathbb{Q}^n$ and $\mathbb{R}^n$; in some abstract sense the former is 'small' (discrete?) while the latter is 'large'. I suspect that ...
0
votes
1answer
96 views

Vertices that create a convex quadrilateral

In how many ways can we choose 4 vertices of a convex n-gon that create a convex quadrilateral (All the inside angles are less than 180) with at least 2 sides of the quadrilateral being sides of the n-...
4
votes
1answer
92 views

relationship between complex numbers

Consider the following: Two equilateral triangles inscribed in a circle. The vertices of the large triangle are the geometric images of the three cubic roots of $z$ (a complex number). The small ...
0
votes
0answers
46 views

Regions of intersected hyperplanes

Let $\mathcal{A}$ be a linear arrangement of hyperplanes in $\mathbb{R}^n$ and $\mathcal{B} := \bigcap_{h \in \mathcal{A}} h$. Show that $ \mathcal{C} := \{ h / \mathcal{B} : h \in \mathcal{A}\}$ ...
0
votes
2answers
54 views

Find the circle touching a line

I have been struggling with this (probably easy to solve) geometry problem for a while. What are the coordinates of the centre and the radius of the circle?