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

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11
votes
2answers
1k views

Relationship between diameter and radius of a point set

Consider a set of $n$ points in $\mathbb{R}^k$. The diameter of this set is the maximum distance between two of its points; its radius is the radius of the smallest (closed) k-ball that contains all ...
11
votes
2answers
1k views

What are all these “visualizations” of the 3-sphere?

a 2-sphere is a normal sphere. A 3-sphere is $$ x^2 + y^2 + z^2 + w^2 = 1 $$ My first question is, why isn't the w coordinate just time? I can plot a 4-d sphere in a symbolic math program and ...
10
votes
6answers
592 views

If $A,B,C,D$ are complex numbers on the unit circle with $A+B+C+D=0$, then they form a rectangle

Let $A, B, C, D$ be points on a unit circle. Prove that if $A+B+C+D=0$, then $A,B,C,D$ make a rectangle. (Use complex numbers.) How do I prove this? I tried to use the dot product of 2 adjacent ...
9
votes
2answers
249 views

Partition of plane into parabolas

The plane is partitioned into parabolas (each point belongs to exactly one parabola). Does it follow that their axes have the same direction?
8
votes
3answers
342 views

$\pi$, Dedekind cuts, trigonometric functions, area of a circle

(I should say at the outset that this question is broad, and may need splitting up. Although I ask several questions, I present them as one because they are not independent of one another, and I am ...
8
votes
2answers
2k views

Numbers of circles around a circle

"When you draw a circle in a plane of radius 1 you can perfectly surround it with 6 other circles of the same radius." BUT when you draw a circle in a plane of radius 1 and try to perfectly surround ...
7
votes
6answers
4k views

A Math function that draws water droplet shape?

I just need a quick reference. What is the function for this kind of shape? Thanks.
6
votes
1answer
2k views

Why does the focus of a rolling parabola trace a catenary?

I keep hearing that when one rolls a parabola on a straight line, the focus traces a catenary. I kept trying to find a proof on the Internet, but no dice. How does one prove this to be true?
4
votes
2answers
4k views

Matrix for rotation around a vector

I'm trying to figure out the general form for the matrix (let's say in $\mathbb R^3$ for simplicity) of a rotation of $\theta$ around an arbitrary vector $v$ passing through the origin (look towards ...
17
votes
4answers
546 views

How fat is a triangle?

The slimness factor of a geometric shape in 2 dimensions is the ratio between the side-length of its smallest containing square and its largest contained square. This is an important factor in ...
10
votes
3answers
10k views

Equation of angle bisector, given the equations of two lines in 2D

I have two lines in 2D expressed with general equation (or implicit equation): First line: $a_1x+b_1y=c_1 \qquad(1)$ Second line: $a_2x+b_2y=c_2 \qquad(2)$ If the two lines are intersecting I will ...
9
votes
3answers
265 views

How to prove $\cos\left(\pi\over7\right)-\cos\left({2\pi}\over7\right)+\cos\left({3\pi}\over7\right)=\cos\left({\pi}\over3 \right)$

Is there an easy way to prove the identity? $$\cos \left ( \frac{\pi}{7} \right ) - \cos \left ( \frac{2\pi}{7} \right ) + \cos \left ( \frac{3\pi}{7} \right ) = \cos \left (\frac{\pi}{3} \right ...
8
votes
2answers
2k views

Divide a triangle into 2 equal area polygons

Through a point outside a triangle, use straightedge and compass to construct a line that divides the triangle into 2 equal areas. (This is my friend's challenge, It was so hard, I don't know where to ...
7
votes
4answers
582 views

Area of intersection between 4 circles centered at the vertices of a square

The centers of four circles are at the vertices of a square of sidelength 100m. Each circle has the radius of 100m. Which is the area of their intersection?
7
votes
2answers
1k views

Formal Proof that area of a rectangle is $ab$

I tried to prove that the area of a rectangle is $ab$ given side lengths $a$ and $b$. The best I can do is the assume the area of a $1\times1$ square is $1$. Then not the number of $1\times1$ squares ...
7
votes
1answer
1k views

Volume of an n-simplex

It's rather tedious to show using Fubini's Theorem and induction on $n$ that the volume of the region $x_1+x_2+...+x_n \leq 1$ with $x_1,...,x_n$ nonnegative is $\frac{1}{n!}$. Is there an easier way ...
6
votes
3answers
12k views

How to determine the arc length of ellipse?

I want to determine the arc length of a ellipse. So what data should I know ? And what law should I use ? For example I have this ellipse on picture below: How can I determine the $d$ length of ...
5
votes
1answer
83 views

Find all such functions defined on the space

$f:\mathbb{R}^3\to \mathbb{R}^{\ast}$ is such that for any non-degenerate tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have : $$f(O)=f(A)f(B)f(C)f(D) $$ Prove that $f(X)=1$ for ...
5
votes
2answers
2k views

Given a width, height and angle of a rectangle, and an allowed final size, determine how large or small it must be to fit into the area

In other words, if I had a rectangle of $10\times 10$ and an angle of $45$, and the allowed area was $100\times 100$, the rectangle would be about $70\times 70$. The allowed area is $100\times 100$ ...
3
votes
4answers
6k views

Common tangent to two circles

Find the equations of the common tangents to the 2 circles: $$(x - 2)^2 + y^2 = 9$$ and $$(x - 5)^2 + (y - 4)^2 = 4.$$ I've tried to set the equation to be $y = ax+b$, substitute this ...
3
votes
1answer
2k views

Can a rectangle be cut into 5 equal non-rectangular pieces?

How to prove that the only figure of which 3 copies can be used to tile a rectangle is a rectangle? Is it possible to cut a rectangle into 5 equal (modulo rotations/reflections) non-rectangular ...
2
votes
2answers
172 views

Proving that $|CA|+|CB|=2|AB|$ in a general $ABC$ triangle

How in this situation (presented in image) can I prove that $|CA|+|CB|=2|AB|$?
1
vote
2answers
161 views

$7$ points inside a circle at equal distances

BdMO 2014 There are $7$ points on a circle.Any 2 consecutive points are at equal distance from one another.How many acute angled triangles can you form taking any 3 of these points? I believe ...
1
vote
2answers
225 views

Finding an unknown angle

Geometry: Auxiliary Lines As shown in the figure:
0
votes
1answer
242 views

Hyperbolic geometry

Post Number: 45 Posted on Friday, 22 March, 2013 - 04:48 pm: I was asked the following question and i do not have any clue on these. Could anyone help me in the beginning of this? Show that there ...
41
votes
14answers
6k views

Do two right triangles with the same long hypotenuse have the same area?

I watched computer monitors and I asked myself, do two monitors with the same display diagonal have the same display area? I managed to find out that the answer is yes, if two right triangles with ...
22
votes
2answers
383 views

About translating subsets of $\Bbb R^2.$

I'm looking for a pair of sets $A,B$ of points in $\Bbb R^2$ such that $A$ is a union of translated (only translations are allowed) copies of $B;$ $B$ is a union of translated copies of $A;$ $A$ is ...
13
votes
6answers
444 views

$\pi$ from the unit circle, $\sqrt 2$ from the unit square but what about $e$? [duplicate]

If one wants to introduce $\pi$ to a not mathematically savvy person, the unit circle would be a good choice. The unit square would be the way to go for $\sqrt 2$. But what about $e$? I've reviewed ...
12
votes
2answers
826 views

$n$ lines cannot divide a plane region into $x$ regions, finding $x$ for $n$

I noticed that $3$ lines cannot divide the plane into $5$ regions (they can divide it into $2,3,4,6$ and $7$ regions). I have a line of reasoning for it, but it seems rather ad-hoc. I also noticed ...
10
votes
3answers
18k views

finding out the area of a triangle if the coordinates of the three vertices are given

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in x-y plane? One approach is to find the length of each side from the coordinates given ...
10
votes
3answers
2k views

The vertices of an equilateral triangle are shrinking towards each other

For an equilateral triangle ABC of side $a$ vertex A is always moving in the direction of vertex B, which is always moving the direction of vertex C, which is always moving in the direction of vertex ...
9
votes
1answer
160 views

Find the area where dog can roam [duplicate]

A dog is tied to circular pillar by a rope. Radius of this pillar is $1m$ and length of rope is $\pi m$. What is an area where dog can roam? I tried to find the area of all semicircles and then to ...
9
votes
3answers
229 views

Cutting a sandwich with a crust

Let $S$ be a simple closed curve in ${\Bbb R}^2$ enclosing a convex region $I$. Must there exist a straight line which cuts $S$ into two pieces of equal length and also cuts $I$ into two regions of ...
8
votes
1answer
92 views

For any three vectors $x,y,z\in\mathbb{R}^d$, we have $ \|y-z\|\cdot\|x\|\leq\|x-y\|\cdot\|z\|+\|z-x\|\cdot\|y\|$

Does anyone know a proof of the following problem? Problem: Show that for any three vectors ${\bf x}, {\bf y}, {\bf z}\in \mathbb{R}^d$ the following holds, $$ \|{\bf y} - {\bf z}\|\cdot \|{\bf x}\| ...
8
votes
2answers
906 views

Is there a Möbius torus?

Does the concept of a Möbius torus make sense: taking a cylinder (instead of a rectangle as in the case of the Möbius strip) and twisting it before joining its ends? Or will the resulting twisted ...
8
votes
3answers
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 ...
7
votes
4answers
2k views

Why does $A^TA=I, \det A=1$ mean $A$ is a rotation matrix?

I know if $A^TA=I$, $A$ is an orthogonal matrix. Orthogonal matrices also contain two different types: if $\det A=1$, $A$ is a rotation matrix; if $\det A=-1$, $A$ is a reflection matrix. My question ...
7
votes
6answers
11k views

Finding a Rotation Transformation from two Coordinate Frames in 3-Space

The question I'm trying to figure out states that I have 3 points P1, P2 and P3 in space. In one frame (Frame A I called it) those points are: Pa1, Pa2 and Pa3, same story for Frame B (namely: Pb1, ...
6
votes
8answers
519 views

Is there a geometrical definition of a tangent line?

Calculus books often give the "secant through two points coming closer together" description to give some intuition for tangent lines. They then say that the tangent line is what the curve "looks ...
6
votes
1answer
503 views

Geodesics on the torus

[This is a follow-up to my question Is there a Möbius torus?] Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus: There are five ...
6
votes
2answers
2k views

Why doesn't a simple mean give the position of a centroid in a polygon?

I was having a look at this question on SO. From what I know, the centroid is the center of mass of an object. so, by definition its position is given by a simple mean of the positions of all the ...
5
votes
1answer
89 views

Is the non-existence of a general quintic formula related to the impossibility of constructing the geometric median for five points?

In particular, in the Computation section of in the Wikipedia page for geometric median, there is this statement: ...but no such formula is known for the geometric median, and it has been shown ...
5
votes
1answer
132 views

prove that line bisect section

There is incircle $\Gamma$ of triangle $ABC$ tangent to $AB,BC,CA$ respectively at $K,L,M$. Point $D$ is the centre of section $MK$. $|DL|$ is diameter of another circle which intersects with $\Gamma$ ...
5
votes
2answers
201 views

Inequality involving sides of a triangle

How can one show that for triangles of sides $a,b,c$ that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} < 2$$ My proof is long winded, which is why I am posting the problem here. Step 1: let ...
4
votes
4answers
686 views

Prove that every isometry on $\mathbb{R}^2$ is bijective

Let $d(x,y)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$ for $x=(x_1,x_2), y=(y_1,y_2)$. A isometry of $\mathbb{R^2}$ is an image $f:\mathbb{R^2}\to\mathbb{R^2}:d(x,y)=d(f(x),f(y))$. Show that every isometry is ...
4
votes
4answers
1k views

Resizing a rectangle to always fit into its unrotated space

(For those coming here looking for answers to rectangle problems it may help to see the related (and solved) question: Given a width, height and angle of a rectangle, and an allowed final size, ...
3
votes
8answers
710 views

Find the approximate center of a circle passing through more than three points

Consider n point $(x_1,y_1), (x_2,y_2),\ldots, (x_n,y_n)$. For $n = 3$ it is easy to find the center of the circle passing through the three points. I wanted find the approximate center of the ...
3
votes
2answers
151 views

Making cuts on a spiral so that all segments are of the same length

The issue we have is this: We have rolls of magnetic strip (about 2 cm in width) and they are rolled in a roll with about 30 windings and about 10m length. The rolls are about 2cm in width as i said. ...
3
votes
1answer
132 views

rectangularizing the square

There is a square that I want to divide to n people, such that each person gets a rectangular piece with an equal area. An obvious option is to cut 1-by-n rectangles of size n-by-1, but the people ...
2
votes
4answers
154 views

Recurrence relation for right-angled triangles stuck-together

Given the image: and that $x_0 = 1, y_0=0$ and $\text{angles} \space θ_i , i = 1, 2, 3, · · ·$ can be arbitrarily picked. How can I derive a recurrence relationship for $x_{n+1}$ and $x_n$? I ...