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

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3
votes
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 ...
2
votes
3answers
56 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 ...
3
votes
0answers
29 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 ...
1
vote
2answers
35 views

Identifying compositions of reflections, and rotations in a hexagon

Let $ABCDEF$ be a regular hexagon that is oriented clockwise (so that a rotation from $A$ to $B$ to $C$ to $D$ to $E$ to $F$ is clockwise). i) Identify $R_{D,120} \circ R_{A,60}$ which are two ...
0
votes
0answers
11 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 ...
1
vote
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 ...
1
vote
0answers
13 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 ...
11
votes
1answer
223 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 ...
1
vote
0answers
54 views

Proof of some inequality on convex sets in $\mathbb{R}^n$

Suppose $\Omega$ is a strictly convex bounded set with smooth boundary in $\mathbb{R}^n$. If $y \in \partial\Omega$, how can we show that $$\langle(x-y),\nu_y\rangle < 0$$ for all $x \in ...
1
vote
1answer
40 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 ...
2
votes
3answers
27 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, ...
0
votes
0answers
21 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 ...
13
votes
6answers
968 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 ...
0
votes
0answers
17 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 ...
-4
votes
0answers
45 views
0
votes
2answers
28 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 ...
1
vote
1answer
90 views

Help with this trigonometry problem?

Is there an easier way of doing this problem: A square tower stands upon a horizontal plane. From a point in this place from which three of its upper corners are visible their angular elevations ...
-3
votes
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 ...
0
votes
2answers
2k views

Optimization of the surface area of a open rectangular box to find the cost of materials

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...
0
votes
1answer
36 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?
1
vote
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 ...
3
votes
1answer
338 views

Geometry - optimal 2D mesh between X expendable points

Say you have X points on a plane. If you connect two points, you form a line. Connecting three points forms a triangle. A line cannot cross a line, and a smaller triangle cannot be created inside a ...
-1
votes
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 ...
1
vote
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 ...
-3
votes
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 ...
0
votes
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 ...
0
votes
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??
0
votes
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 ...
4
votes
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 ...
0
votes
0answers
28 views

How to calculate the volume between points? [on hold]

Say we have a set of finite points: $$ A=\{ (x_1,y_1,z_1), (x_2,y_2,z_2) , \dotsc , (x_n,y_n,z_n) \} $$ Assuming these are surface points of a 3D shape. I want to calculate the volume confined ...
-2
votes
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 $AB^2+BC^2+CA^2=3(GA^2+GB^2+GC^2)$
1
vote
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?
1
vote
1answer
31 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 ...
3
votes
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 : $$ \sin x +\sin y +\sin z\le \frac{3}{2} $$
2
votes
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 ...
4
votes
2answers
53 views

does this convex set have a specific name?

Let $x_1,\dots,x_N$ be points of $\mathbb{R}^n$. Define the following set: $\mathcal{A} = \left\{\sum_{j=1}^N a_j x_j : -1 \le a_j \le 1, \, \, \forall j=1,...,N\right\}$. It is an easy exercise to ...
7
votes
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 ...
0
votes
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?
2
votes
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 ...
7
votes
2answers
6k views

Why does every direction at the north pole point south?

Why does every direction at the north pole point south? Why doesn't this happen at any other point on (face of the) earth? Is this due to convention used by humans or is there a geometrical ...
1
vote
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 ...
8
votes
4answers
12k views

The intersection of two parallel lines

My teacher told me that two parallel lines have a point of intersection, it is called Point at infinity. But I I can't understand how it can be true or how was it proved, can someone explain this to ...
0
votes
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.
2
votes
3answers
1k views

Proving a property of an ellipse and a tangent line of the ellipse

Suppose that there is line $l$ that is tangent to an ellipse $A$ at point $\,P\,$. The ellipse has the foci $F'$ and $F$. One then creates two lines - each from each focus to the tangency point ...
2
votes
0answers
70 views

Problem with an inclined cone and planes

From the image given below, I want to prove that there exists a unique plane $p \neq P$ s.t. $p \cap$ inclined cone $=$ circle centered at $O_{2}$. I also want to prove that if ray $SO_{1}$ (where ...
34
votes
3answers
2k views

Fascinating Lampshade Geometry

Today, I encountered a rather fascinating problem in a waiting room: Notice how the light is being cast on the wall? There is a curve that defines the boundary between light and shadow. In my ...
0
votes
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 ...
1
vote
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}}} $?
10
votes
4answers
302 views

An ancient Japanese geometry problem.

NOTE: This very difficult problem of elementary geometry has an ancient Japanese source (See “Sacred Mathematics: Japanese Temple Geometry”. Princeton University Press, 2008, by F. Hidetoshi & ...
1
vote
1answer
30 views

Relative distance to rotated object

An object (with center $O_{2}$) has been rotated by an angle $\alpha$. There are two images of the object taken by a camera (centered at $O_{1}$), and two points $x_{1}$ and $x_{2}$ that are actually ...