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
votes
2answers
33 views

Prove the existence of minimal height of a convex polygon

Suppose we have a polygon in $\mathbb{R}^n$. Obviously, we can always trap the polygon into two parallel hyperplanes perpendicular to any given direction. Now the question is, how to prove the ...
0
votes
1answer
39 views

Cutting the Plane

Into how many parts at most is a plane cut by $n$ lines? Into how many parts is space divided by $n$ planes in general position? My approach: $$p(n+1)=p(n)+n+1$$ $$s(n+1)=s(n)+p(n)$$ This solution ...
4
votes
1answer
23 views

constructing a spherical triangle using only the laws of sines and cosines

I have a spherical triangle with corners $A,B,C$, angles $\alpha, \beta, \gamma$ and sides $a,b,c$ (which are opposite to the corresponding corners/angles). I am given $a,b$ (with $a>b$) and ...
0
votes
4answers
52 views

Geometry problem, find lengths

Here's my problem: I know the angles $\alpha$ and $\beta$, and the line length $a + b$. I need to find the lengths $a$ and $b$ and the height of the triangle. I came up with the identity ...
0
votes
1answer
50 views

Another olympiad question related to External principle (regarding geometry problem)

Into how many parts at most is a plane cut by $n$ lines? (b) Into how many parts is space divided by $n$ planes in general position First i was thinking about the approach (not able to find it). ...
1
vote
1answer
23 views

Circular cross-sections characterize spheres

The intersection of a set $A \subset \mathbb{R^3}$ with all planes is always a circle. Prove that $A$ is a sphere. We regard single points and $\emptyset$ as degenerate circles & spheres. I ...
0
votes
1answer
27 views

Find the Distance Point to Line with Point on Line and Direction Vector

We are given two Vector3's for the Line; Vector3 Point on the Line; Vector3 Direction along the line. We are also given the certain point which is the point outside the line. So how am i supposed to ...
3
votes
3answers
82 views

How is $\sin 45^\circ=\frac{1}{\sqrt 2}$?

I've been reading about the proof of $\sin 45^\circ=\dfrac{1}{\sqrt 2}$ in my book. They did it as following, let $\triangle ABC$ be an isosceles triangle as shown, Since the triangle is isosceles ...
2
votes
1answer
27 views

$U$ bisected by all hyperplanes $\implies$ $U$ symmetric?

Let $U \subset \mathbb{R}^n$ be a bounded open convex set, such that every hyperplane passing through the origin divides $U$ into two sets of equal volume ($n$-dimensional Lebesgue measure?). ...
2
votes
1answer
24 views

Analytic geometry line segments

This is a very interesting analytic geometry math problem that I came across in an old textbook of mine. It is quite nice and I decided I would share it with MSE for future reference and a fun time?! ...
-2
votes
2answers
41 views

Elementary problem in geometry [closed]

The problem asks to find the angle at $C$. The distance between $A$ and $B$ is $12 \space m$ and the distance between $B$ and $C$ is $8\space m$. Anyone got an idea?
0
votes
1answer
12 views

In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base.

I have the following theorem : "In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base." (Figure is in the link) ...
0
votes
0answers
22 views

A length of fence encloses an area alongside a river — what is the optimal shape to maximize area? [duplicate]

You have $100$ meters of fence . There is a perfectly straight riverbank, much longer than $100$ meters, so you have plenty of room to work with. What is the optimal shape and dimensions that ...
1
vote
1answer
31 views

Show by definition that $M=\{(x,y,z)|36x^2+4y^2-9z^2=36\}$ is a surface in $\Bbb R^3$

Show by definition that $M=\{(x,y,z)|36x^2+4y^2-9z^2=36\}$ is a surface in $\Bbb R^3$. Definition A surface in $\Bbb R^3$ is a subset $M$ of $R^3$ such that for each point $p$ of $M$ there exists a ...
5
votes
1answer
85 views

Curve meeting itself everywhere

(related, but not a duplicate: curve which crosses itself at every point ) When reading the comments to the question above, it has been pointed out that if by "cross" we mean that for every ...
0
votes
4answers
46 views

Very elementary question on isosceles triangles, Geometry

Suppose we have an isosceles triangle with sides $l,l,k$. Is it a definition that we always must have that $l \geq k $ or this need to be proved ? MY thought on how to prove this: Call $\alpha$ the ...
2
votes
3answers
42 views

Is this equation a parabola or a hyperbola?

In a 1972 paper by Robert Merton, the following equation is derived: $$\sigma(\mu;A,B,C,D)=\sqrt{\frac{A \mu^2-2B\mu+C}{D}}$$ This is known as the Markowitz frontier in finance. When this is ...
0
votes
2answers
19 views

How to check if a point is in the direction of the normal of a plane?

I have a plane, defined by a normal vector $n$ and a point $p$. I also have a point $a = (x, y, z)$. Based on this information, how do I know if the point $a$ exists somewhere past the plane in the ...
1
vote
3answers
63 views

How to “rotate” points through 90 degree?

I am trying to do some intersection tests and so the math gets weird if two certain points have the same $x$ coordinate and so infinite slope. The points can be anywhere in any quadrant. I want to ...
2
votes
1answer
77 views

Prove: $m$ balls in $\mathbb{R^3}$ cut $\mathbb{R^3}$ into less than $m^3$ connected components.

I need to prove or at least to understand why $m$ balls in $\mathbb{R^3}$ cut $\mathbb{R^3}$ into less than $m^3$ connected components. But I've no idea how to deal with it. I even tried to draw it ...
3
votes
1answer
57 views

Finding the closest point to a set of lines in 2D

I would need to write an algorithm to find the closest point to a set of lines. These lines are infinite and are not parallel between each other. Closest point means that point where the sum of the ...
4
votes
2answers
47 views

How would you call geometric objects that lie on a single surface, e.g. a sphere, plane, torus, etc.

I'm looking for an extension of the name coplanar to something like "cosurfacial", but I guess their must be a correct term.. Edit: In the comments, the context was asked for where I would use that ...
1
vote
0answers
24 views

Horn angles and Euclid's elements.

We have the following statement by Euclid : "I say further that the angle of the semicircle contained by the straight line BA and the circumference CHA is greater than any acute rectilinear angle, and ...
0
votes
0answers
31 views

Proving a quadrilateral is an isosceles trapezoid

Warning: You'll probably need pencil and paper to follow this. Recently I came across the following problem in a middle/high school geometry textbook: $\ast$ Suppose $\ QUAD\ $ is a quadrilateral ...
0
votes
0answers
22 views

Realize a group as a subset of a metric space

I have a question about the definition of "realize". I read the following definition: A group $G$ is said to be realized as a subset $Y \subseteq X$ of a metric space $X$ if the isometry group ...
0
votes
1answer
19 views

Is the path with the highest average value the same as the path with the lowest total difference from the function's maximum?

If you have a function $f(x, y)$ and you draw two paths (curved lines) from points A to B where: The first path is the path with the highest average value (if the value at distance $d$ from the ...
2
votes
0answers
49 views

Is the following a conic section

All vectors are in $\mathbb{R}^3$ and only $\mathbf{r} = \left[ x; y; z \right]$ is unknown. My question is does the following system define a conic section in the $x-y$ plane and, if so, how can I ...
1
vote
1answer
33 views

Euclid's elements proposition 15 book 3

http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII15.html I have understood the proof in general. It is only a small detail which i'm not sure. Maybe it's because english isn't my first ...
1
vote
4answers
45 views

Compute the area of specific shapes [closed]

I'm trying to calculate the dashed area in the following pictures, and I can't solve them. I tried to guess the areas, subtract some shapes from others, but I'm confused if I calculated them wrong or ...
3
votes
2answers
61 views

Centroid of a Triangle on a inscribed circle

$AB$ is the hypotenuse of the right $\Delta ABC$ and $AB = 1$. Given that the centroid of the triangle $G$ lies on the incircle of $\Delta ABC$, what is the perimeter of the triangle?
1
vote
0answers
54 views

Calculating the Area of a Circle Occupied by a Rectangle

This is a question regarding how to calculate the area of a circle occupied by a rectangle when that rectangle is larger than the circle (see this link for a example image ...
1
vote
1answer
12 views

Proof of Specific Distribute Property for Vectors

Wasn't really able to find something here or on Google which answers my question. I am asked to prove the distributive property of vectors such that $$(r + s) * \vec{a} = r * \vec{a} + s * \vec{a}$$ ...
1
vote
0answers
33 views

Solve for Sides of a 5-Sided Irregular Polygon

I have a 5-sided irregular polygon and I know the lengths of 4 of its 5 sides and 2 of its 5 angles. Is there a way to know the length of the 5th side using this information?
4
votes
5answers
145 views

What fraction of a sphere can an external observer see?

Here is a geometry problem. Let there be a ball of radius R and let's call it the Moon. Let there be an external observer: A. A is at a distance d to (the surface of) the Moon. [Edit] A is a ...
0
votes
1answer
14 views

“How to sort vertices of a polygon in counter clockwise order?”: Computing Angle?

my question relates to the answer to the following question: How to sort vertices of a polygon in counter clockwise order? I don't have a strong background in linear algebra... I don't understand ...
0
votes
1answer
31 views

Tetrahedron subdivision

What are all the possible subdivisions of the P3 tetrahedron (i.e. for each face, 3 vertices plus two points per edge, located at 1/3 and 2/3, and the centroïd of the face, so a total of 20 points for ...
0
votes
1answer
32 views

Points in a given volume/Area

I have a rectangular prism(3D bounding box) for which i have the point(i.e center of gravity) and the height,width,depth dimensions . Given these parameters, is it possible to find all the points that ...
2
votes
1answer
42 views

Prove the Median of a Trapezoid Bisects Both Diagonals

I am trying to prove that the median of a trapezoid bisects both of the trapezoid's diagonals using basic vector operations (addition, subtraction, etc.). I have tried to do this by labeling the two ...
0
votes
2answers
44 views

calculate the volume

There is a triangular prism with infinite height. It has three edges parallel to z-axis, each passing through points $(0, 0, 0)$, $(3, 0, 0)$ and $(2, 1, 0)$ respectively. Calculate the volume within ...
-1
votes
0answers
50 views

Line of Irrational Length? [duplicate]

If we drew a line of irrational length using pythagorean theorem, then is the length of the line really irrational? Can line of irrational length really exist? Will it be possible for a computer with ...
0
votes
0answers
20 views

Upper Bound on Vector Length

I read the following statement recently: Note that for any vector $a=(a^1,a^2,\cdots, a^n)$ we have $|a|\leq|(a^{i_1},a^{i_2},\cdots,a^{i_k})|$ for any choice of $i$. I have no idea what this ...
0
votes
1answer
32 views

Euclid's elements proposition 13 book 3

"Then, since on the circumference of each of the circles ABDC and ACK two points A and C have been taken at random, the straight line joining the points falls within each circle, but it fell within ...
0
votes
1answer
34 views

How can I create a level and perfect east-west line?

Imagine you have a 10'x30' solar array. (A) = 30' Side A needs to be positioned perpendicular or facing True South, which is 14 degrees west of magnetic south. (Look at S on compass... 180 degrees. ...
1
vote
1answer
31 views

Exploiting geometric invariants via group theory

Let $T$ be the set of all plane triangles. The problem is to find $t \in T$ s.t.h. a predicate $P(t)$ holds. At present, I'm doing this by a form of randomized search procedure (effectively via a ...
1
vote
2answers
42 views

Can ANY 2 or 3 dimensional shape be reversed engineered to give an equation (formula) for its shape??

Can ANY 2 or 3 dimensional shape be reversed engineered to give an equation (formula) for its shape? In other words given ANY 2 or 3 dimensional shape that ones draws on a graph can one reverse ...
2
votes
2answers
54 views

Proving that every patch in a surface $M$ in $R^3$ is proper.

Problem Prove that if $\mathbf{y}:E\to M$ is a proper patch, then $\mathbf{y}$ carries open sets in $E$ to open sets in $M$. Deduce that if $\mathbf{x}:D \to M$ is an arbitrary patch, then the image ...
1
vote
3answers
37 views

Find the magnitude of theta

I was given a problem telling me to find the magnitude of $\theta$. I have been trying to reason it out using complements and such, but I haven't been able to get it. The diagram seems strange to ...
0
votes
0answers
20 views

Transpose is just the way of generalizing a dot product?

It seems like $a^Tb$ is the same as writing $a \cdot b$ in matrix form. 1) Why is $n \times 1$ and $n \times 1$ matrix multiplication undefined? 2) Is this just a generalization of the dot ...
0
votes
0answers
36 views

BMO1 2005/06 Question 5 Geometry Problem

Let $G$ be a convex quadrilateral. Show that there is a point $X$ in the plane of $G$ with the property that every straight line through $X$ divides $G$ into two regions of equal area if and only if ...
3
votes
1answer
68 views

Translation from schemes to varieties

At the moment, I know very little algebraic geometry (sadly!) so I apologise for the silliness/stupidity of these questions. Set up: Let $k$ be a field (not necessarily algebraically closed). Take ...