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

learn more… | top users | synonyms

13
votes
4answers
2k views

What is a point?

In geometry, what is a point? I have seen Euclid's definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know ...
4
votes
1answer
1k views

the intersection of n disks/circles

Given that $n$ disks/circles share a common area, meaning that every two of them intersect or overlap one another, and we know their coordinates $(x_{1},y_{1},r_{1})$, $(x_{2},y_{2},r_{2})$, ..., ...
24
votes
1answer
2k views

if there are 5 points on a sphere then 4 of them belong to a half-sphere.

If there are 5 points on the surface of a sphere then there is a closed half sphere containing at least 4 of them. It's in a pidgeonhole list of problems, but I think I have to use rotations in more ...
24
votes
5answers
956 views

What is (a) geometry?

There is no question what topology is and what it's about: it's about topologies (= topological spaces), and that's it. There is also no question what (universal) algebra is and what it's about. ...
8
votes
3answers
1k views

Proving the area of a square and the required axioms

I recently realized the area formula of all polygons, and most basic figures can be proven from the areas of square and rectangle. For example if we know the area of rectangle, we can the area formula ...
8
votes
2answers
335 views

On the problem 1 of Putnam 2009

(This is adapted from problem 1 of Putnam 2009) Find all values of $n$ such that the following is true: There is a non-constant function $f : \Bbb{R}^2 \longrightarrow \Bbb{Z}_2$ such that for any ...
7
votes
1answer
206 views

How can we draw a line?

We want to join by a line two distinct points $A$ and $B$. We have only a ruler of length $l>0$ and a pen. If $AB>l$ how can we do this?
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 ...
11
votes
5answers
3k views

Pythagorean Theorem Proof Without Words 6

Your punishment for awarding me a "Nice Question" badge for my last question is that I'm going to post another one from Proofs without Words. How does the attached figure prove the Pythagorean ...
10
votes
6answers
640 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
253 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
348 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
3k 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
3k 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 ...
41
votes
4answers
3k views

Do circles divide the plane into more regions than lines?

In this post it is mentioned that $n$ straight lines can divide the plane into a maximum number of $(n^{2}+n+2)/2$ different regions. What happens if we use circles instead of lines? That is, what ...
17
votes
4answers
565 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 ...
14
votes
2answers
3k views

Trisect unknown angle using pencil, straight edge & compass; Prove validity of technique

This question was posed by my high school geometry teacher, for extra credit: Is it possible, using only a pencil, a straightedge (not a ruler) and a compass to trisect an angle of unknown value? ...
13
votes
2answers
833 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
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
255 views

Beginning of Romance

I am a 17-year old student in India, in the standard 12th grade. Recently, I found the fascination in mathematics, and I am eager to dig in further. Currently, the only textbooks I have are the ones ...
9
votes
3answers
272 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
635 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
13k 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
85 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 $100\times 100$ ...
4
votes
1answer
1k views

Is a sphere a closed set?

The unit sphere in $\mathbb{R}^3$ is $\{(x,y,z) : x^2 + y^2 + z^2 = 1 \}$. I always hear people say that this is closed and that it has no boundary. But isn't every point on the sphere a boundary ...
3
votes
4answers
7k 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
173 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
172 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
247 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
386 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
452 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 ...
11
votes
4answers
2k views

What are affine spaces for?

I'm studying affine spaces but I can't understand what they are for. Could you explain them to me? Why are they important, and when are they used? Thanks a lot.
11
votes
4answers
5k views

Equilateral triangle whose vertices are lattice points?

Is it possible to construct an equilateral triangle with vertices on lattice points? I think the answer is no, but how can I prove this? I started with a triangle with coordinates $(0,0)$ $(a,b)$ ...
10
votes
3answers
21k 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
164 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
232 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
946 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
4answers
13k 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 ...