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

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5 views

Spherical Lines on the Complex Plane

Say you consider the stereographic projection of the unit sphere onto the extended complex plane. If you have 2 points on the plane, neither corresponding to the north pole, the spherical line ...
5
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0answers
11 views

Reference Request: Complete Proof of Braikenridge–Maclaurin Theorem

Where can I find a reference to a complete proof of the Braikenridge–Maclaurin theorem, which is stated as: If the three pairs of opposite sides of (an irregular) hexagon meet at three collinear ...
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0answers
13 views

How to parametrize circles on a sphere by the distortion of the equator?

I guess am having a very silly problem right now. Considering a unit sphere $S^2$ and, for example, a curve, in spherical coordinates, $c(t)=(1, \frac{\pi}{2},t)$ that goes around the equator how can ...
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0answers
19 views

Interpretation of median length for an invalid triangle

Background: My very first and naive take on the Project Euler problem 513 went wrong, as I counted also triples violating the triangle inequality. Many formulas return an invalid result for an ...
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0answers
16 views

Proof of Compound Angle from Ptolemy's Theorem

I have a query regarding a proof I'm reading on the additive Sine compound angle formula, which uses Ptolemy's theorem. http://www.cut-the-knot.org/proofs/sine_cosine.shtml I'm looking at the ...
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4answers
104 views

Technique for proving four points to be concyclic

While making my way through an exercise, I stalled on question 7: 7. Prove that the points $(9, 6)$, $(4, -4)$, $(1, -2)$, $(0, 0)$ are concyclic. The book does not provide any guidance on how ...
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3answers
28 views

Find all line equations that are tangent to $x^3 - x$ and pass through $(-2,2)$

So I have the equation: $f(x) = x^3 - x$ So we know that the slope of the curve for some $x$ is given by: $f'(x) = 3x^2 - 1$ And need to find equations of lines that are tangent to that curve, ...
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1answer
288 views

Can I represent groups geometrically?

I have just taken up abstract algebra for my college and my professor was giving me an introduction to groups, but since I like geometric definitions or ways of looking at stuff, I kept thinking, "How ...
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0answers
22 views

Geometric solution for $\frac{1-R}{| 1 - R e^{j\theta} |} = k$

Given: $\frac{1-R}{| 1 - R e^{j\theta} |} = k$ How to solve for R? (Suppose R is the only unknown quantity -- the task is to rearrange with R as subject). I encountered this problem in an academic ...
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45 views

Mathematics doubt. [on hold]

What is mathematics. I need a proper definition mathematics.
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0answers
18 views

Specific polygons in \R^{3} [on hold]

Find all (convex) polyhedra P in $\mathbb{R}^{3}$ with following property: for every two vertices $v, u \in P$: $[u,v] \in \partial P$ . Here $[u, v] = \{w \; \vert \; w = tv + (1-t)u, \; t \in ...
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0answers
34 views

Projective line intersecting 3 projective subspaces

I am trying to solve the following problem: Let $\mathbb{P}(U),$ $\mathbb{P}(V)$ and $\mathbb{P}(W)$ be projective subspaces of dimension $k,$ $l$ and $m$ respectively in $\mathbb{P}_K^n$. Suppose ...
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2answers
29 views

AMC: Triangle area problem

In rectangle ABCD below, points F and G lie on segment AB such that AF = FG = GB and E is the midpoint of segment DC. Also, segment AC intersects segment EF at H and segment EG at J. The area of ...
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2answers
25 views

How find the length of the third of the chord for of a circle of radius r? [on hold]

Three chord of a circle of radius $r$ are the sides of the triangle inscribed in the circle. Two of these chords have a length $\frac{r}{2}$ and $r\sqrt{3}$. How find the length of the third of the ...
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0answers
6 views

Identify the combination formed by first applying the glide reflection $γ_{A,B}$ and then applying $γ_{C,D}$

Question: Identify the combination formed by first applying the glide reflection $γ_{A,B}$ and then applying $γ_{C,D}$ where $A = (0, 0), B = (2, 0), C = (1, −2), D = (1, 0)$. Before, I jump in ...
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2answers
47 views

Surface area of a section of the unit sphere

Let $v$ be a vector on the unit sphere in $\mathbb{R}^n$ and let $S(\epsilon)$ be the set of vectors $s$ on the same sphere such that $$ |s \cdot v| \leq \epsilon.$$ What is the surface area of ...
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6answers
997 views

Can someone explain 4th dimensional objects?

I'm not sure if I should ask this in mathematics or in physics. From what I can tell, there are only 3 dimensions: X, Y, and Z. However, I have seen a lot of things about fourth and even fifth ...
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1answer
37 views

Tangent lines of a smooth curve $C \subseteq \mathbb{P}^2$

Let $C$ be a smooth curve, given as the zero locus of a homogeneous polynomial $f \in \mathbb{K}[x_0,x_1,x_2]$. Consider the morphism $\varphi_C:C\rightarrow\mathbb{P}^2$ such that ...
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0answers
9 views

Finding closest rectangle to another using concept of closest edge [on hold]

I know coordinates and size of rectangles. My goal is to find 'the closest' rectangle to one special rectangle using the concept of closest edge and also to find distance between them??
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1answer
10 views

A question on the requirement of a quadrilateral being an adventitious quadrangle

There is a special type of problem called Langley’s Adventitious Angles. See http://en.wikipedia.org/wiki/Langley%E2%80%99s_Adventitious_Angles The problem was solved and has the following ...
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3answers
30 views

Calculating last two angles of a quadrilateral when two angles and (relative) side lengths are known.

I've recently run into a problem that I can't seem to get. Given a quadrilateral, $ABCD$, with angle $D$ equaling 54 deg, and with the length of $BC$ equaling $CD$ while being double the length of ...
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0answers
19 views

Find out the vertices of a regular polygon, given one side and number of sides [on hold]

We are given: 1)one side of the regular polygon , $ (x_1,y_1) $ and $ (x_2,y_2) $ 2)number of sides of the polygon 3)construct the polygon in counterclockwise direction And are asked to find all ...
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2answers
36 views

In a $\Delta ABC$, G is the centroid. prove that [on hold]

In a $\Delta ABC$, G is the centroid.Prove that $AB^2+BC^2+CA^2=3(GA^2+GB^2+GC^2)$
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2answers
35 views

How to calculate line-line distance when cross product of directions is 0?

I have the lines $$\frac{x-1}{2} = 1-y = \frac{z-2}{3} \tag{1}$$ and $$\frac{x+1}{4} = \frac{4-y}{2} = \frac{z+1}{6} \tag{2}$$ I want to compute the distance between them. I started by putting ...
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1answer
32 views

Why is this a convex polygon?

Let $\text{E}(2)$ be the group of isometries of the plane $\mathbb R^2$. Then $\text{E}(2)=\text{O}(2)\times\mathbb R^2$ as a topological space and is the semi-direct product as groups. Let $G$ be a ...
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0answers
36 views

definition of largest circle containing in a convex body

For a convex body B(compact convex set with non-empty interior) what does each of these following values mean? 1- $$\ \sup_{x\in B}\inf_{y\in cB} d(x,y) $$ and 2- $$\ \sup_{x\in B}\inf_{y\in ...
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2answers
17 views

Calculating the currvature of a tractrix

I'm trying to find the curvature of a tractrix expressed in the form $r(t)=(\sin{t},\cos{t}+ln(\tan{(\frac{t}{2}))} $. From what I've found on the Internet it appears that people arrive at the ...
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1answer
21 views

will the ladder reach the height of a 3.5m window.

A ladder of length 4m leans against the wall of a house. The foot of the ladder is 2m from the wall. Will the ladder reach a window 3.5m high?
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0answers
27 views

Point in four dimensions

To describe a point in $3$D: Three parameters $ r,\phi, \theta $ are used in spherical coordinate system. Taking them pairwise,two as $(r, \theta)$ in azimuth plane and two $(r,\phi )$ in meridian ...
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2answers
379 views

Is a circle classified as an ellipse?

I read that an ellipse had $2$ focal points. So, I thought if a circle had $2$ points that were simply infinitesimally close together wouldn't it be classified as an ellipse? Help would be ...
3
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0answers
53 views

How prove that : $ \sin x+\sin y+\sin z \le \frac{3}{2} $ for triangle $ABC$

Let $ G $ be the centroid of $ \triangle ABC $ , such that $ \angle{GAB}=x,\angle{GBC}=y,\angle{GCA}=z $, How do I prove that : $$ \sin x +\sin y +\sin z\le \frac{3}{2} $$
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1answer
37 views

Draw the line segment joining the centers of two circles. Where does it meet the circles?

I'm trying to construct a line segment between two circles. Given each radius and $x$, $y$ center of each circle, how can I find the endpoints for the blue line segment?
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0answers
20 views

What are some techniques of specifying a molecules structure using the least amount of information?

For instance say I have a water molecule I can describe it's structure by two bond lengths and a bond angle. Are there any neat math tricks or representations of objects that I could use to describe ...
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1answer
35 views

Curvature of plane curves

What is the neatest way to derive the formula for the curvature $\kappa =\frac{||y'x''-y''x'||}{(x'^2+y'^2)^{\frac{3}{2}}} $?
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0answers
16 views

Signed unit normal

I'm trying to study for one of my exams and the past papers have no solutions. I had to define the signed unit normal and the signed curvature. The signed curvature, $\kappa_s$ being such that ...
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2answers
62 views

Probability of triangle to be acute?

Suppose that someone randomly picks $3$ points $A, B$ and $C$ on a fixed circle. What is the probability of triangle $ABC$ to be acute?
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0answers
15 views

How do I find all lines of $\mathbb{P^{3}}$ meeting $L, M$ and $N$?

Let $L, M$ and $N$ be $3$ lines of $\mathbb{P^{3}_{R}}$ such that $$L \cap N = \emptyset, \quad M \cap N = \emptyset \quad \text{and} \quad L \cap M = \{ G \} \ \ \text{(one point)}.$$ I'm asked to ...
3
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1answer
18 views

Differential length of a logarithmic spiral

I am working on a problem that asks to find the magnetic field at the origin of a logarithmic spiral $r = e^\theta$ from $\theta = 0$ to $\theta = 2\pi$, where the angle $\theta$ is measured ...
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2answers
51 views

finding angle value inside this triangle

I need a method to calculate the angle X in the image below, I know its value (30 degree) but how ?!! thank you.
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0answers
35 views

Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$

Let $P_1 P_2 \dots P_n$ be a convex polygon in the plane. Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$ Show ...
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3answers
35 views

Why do lines in the poincare model meet the infinite edge at right angles?

I know the lines are generated by projecting geodesics on a hyperboloid to a plane and the boundary of the disk comes from the asymptotic cone around the hyperboloid, but I just don't see why the ...
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1answer
16 views

What are the images of a point in a multi-layer mirror systems?

We all know that there are infinite images of a point which is located between two parallel mirrors. Also, the locations of the images can be easily obtained. Generally, how to locate the images of a ...
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1answer
21 views

Let the diameter of a subset S of the plane be defined as the maximum of the distance between arbitrary of points of S

Let the diameter of a subset S of the plane be defined as the maximum of the distance between arbitrary of points of S. Let$$S=\{(x,y):(y-x)\le0,(x+y)\ge0,x^2+y^2\le2\}$$ Then the diameter of S is ...
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1answer
23 views

Given sides and a bisection, find angles in a triangle

Consider a triangle $ABC$ where the angle $A$ is $60^{\circ}$. Draw its bisection intersecting $BC$ at $D$. Let $AB = x$, $BD = y$ and $AC=x+y$, $\angle ABC = \alpha$ and $\angle ACB = \beta$. Find ...
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0answers
13 views

Boundedness condition of Minkowski's Theorem

Statement: "Let L be a lattice in $R^n$ and $S\subset R^n$ be a convex, bounded set symmetric about the origin. If $Volume(S) > 2^ndet(L)$, then S contains a nonzero lattice vector. Moreover, if ...
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1answer
32 views

“Reverse engineering” of a geometric illustration

The following enigmatic illustration can be found here, unfortunately without any accompanied comment or short description: Can you deduce its meaning? What was the way it was constructed?
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2answers
46 views

Complicated Planar Geometry [duplicate]

A regular polygon $\mathcal{P}$ is inscribed in a circle $\Gamma$. Let $A$, $B$, and $C$ be three consecutive vertices of the polygon $\mathcal{P}$, and let $M$ be a point on the arc $AC$ of $\Gamma$ ...
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0answers
15 views

Find out rotation and angle given a square

Consider I have a camera and a square, the projection of the square on the camera would give me a "distorted" square. Can I possibly find out the angle of the camera relative to the square and the ...
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1answer
14 views

what is the difference between an elliptical and circular paraboloid? (3D)

My textbook uses the terms interchangably, and they look the same in graphs, so I was wondering if there a difference between the two? Thanks!
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2answers
34 views

How to reverse this mathematical formula

I am using the following function to compute a "Y" value from a Latitude ...