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
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1answer
537 views

Validity of proof for surface area of a sphere [duplicate]

On a geometry test I forgot the formula for the surface area of a sphere so I derived it and ended up being right. But it seems like my derivation is wrong. I got the surface area formula by taking ...
0
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0answers
132 views

Find the area and the measure of interior angles of the rhombus $ABCD$

I have such a problem which I am really earnest to solve, but I got stuck and hope you'll help me to find a solution: So we have a rhombus $ABCD$(in it a circle is inscribed) in which ,we see from ...
0
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1answer
48 views

Non constant function of two points invariant under Affine transformation proof

Here is the question; Prove that there does not exist any nonconstant function of pairs of distinct points $P,Q\in\mathbb{R}^2$ or of triples of distinct non collinear points $P,Q,R\in\mathbb{R}^2$ ...
4
votes
3answers
98 views

Nice parameterization of $x^2 + y^2 - kx^2y^2 =1$

Can anyone find a nice simple parameterization of this curve. Just the quarter where $x \ge0$ and $y \ge0$ would be fine. The parameterization should be "nice" in the sense that the first derivative ...
0
votes
1answer
191 views

Radius of a circle (partially) inscribed off-centre in a square

Suppose we have a square diamond. There is a circle, where the top point of the circle is at the top corner of the diamond, and it touches the bottom two sides of the diamond. What is the radius of ...
5
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2answers
151 views

Geometry after Khan academy's tutorials

I always liked geometry at school, so by the side of my normal studies I've been going through the Khan academy videos. Could anyone suggest some good books that takes these geometry topics further? ...
1
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0answers
65 views

The Least Area For a Needle to Pass Through a Curve?

I don't know if this question is a famous one. One of my fellows asked me these questions to tease me, but I was able to find a solution for only one of these: There is one needle of length $2$ ...
1
vote
2answers
916 views

Finding a normal to an ellipsoid

Let $E$ be an ellipsoid centered at $v = (x,y,z) \in \mathbb{R}^3$ and let $T:\mathbb{R}^3 \to \mathbb{R}^3 $ be a linear transformation which transforms $E$ to a sphere $S$ with a radius of length ...
0
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1answer
38 views

Trying to revise a formula I was once given. How many rectangular prisms are in a $n \times n \times n$ cube?

I post it the other day. The only answer I got is that the total number of rectangular prisms in a cube is equal to ${n+1 \choose 2}^3$. But using $n=2$, I found the formula to be wrong. When counting ...
1
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0answers
159 views

Lateral area of a cone with oblique base (elliptical base)

Can someone help me find the lateral area of a straight circular cone with oblique base ( inclined elliptical base) as seen at the following link. ...
0
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1answer
82 views

Generalised Pythagorean Theorem?

$|a+b|^2=|a|^2+|b|^2+2 Re(\overline ab)$ Can anyone explain this equality to me? How it is derived?
0
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1answer
83 views

Converting 3D into 2D

I have a quad and I'm trying to convert its vertices so that they're facing the camera which is lying at 0,0,1 looking down the Z, or not even specifically facing the camera, just so they're facing up ...
5
votes
2answers
556 views

Why do folded concentric circles and rectangles form a hyperbolic paraboloid?

Here is a "self-forming" origami that I made from folding concentric circles - it would also happen if I folded concentric rectangles. How can the fold shapes such a saddle-like geometry?
0
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0answers
51 views

Prove equality of angles in parallelogram

In a parallelogram $ABCD$, $P$ is a point inside the parallelogram such that $\angle APB + \angle CPD=180$. Prove that $\angle PDC=\angle PBC$.
5
votes
2answers
91 views

How to characterize rotations in $\mathbb{R}^n$?

I am studying the performance of an optimizer algorithm to find the $$ \textrm{argmin}_{x\in \mathbb{R}^n} f(x) \text{ where } f : \mathbb{R}^n \rightarrow \mathbb{R} $$ I would like to test how the ...
2
votes
3answers
75 views

Complex Numbers of Unit Modulus

if $z_1$, $z_2$ and $z_3$ are Complex Numbers of Unit Modulus Such That: \begin{equation} |z_1-z_2|^2+|z_1-z_3|^2=4 \tag{1} \end{equation} Find the value of $$|z_2+z_3|$$
1
vote
1answer
242 views

Proof: Convex set of a quadrilateral is a convex quadrilateral

Prove that $\Box ABCD$ is a convex set whenever $\Box ABCD$ is a convex quadrilateral. Things I know: A set of points $S$ is said to be a convex set if for every pair of points $A$ and $B$ in $S$, ...
1
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4answers
55 views

Perpendicular versus perpendicular bisector

We have $AH=HB$ and $BG=GC$ in the image below. Why is $AD=2\times FG$?
0
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1answer
35 views

3d analogue of an ellipse

An ellipse is the set of all points X in 2d space such that the the sum of the distances AX and BX is a given constant constant, where A and B are given points. What is the name for the set of all ...
5
votes
2answers
358 views

Relationship between circles touching incircle

I am trying to derive a relation between radius of those outer circles and radius of the incircle. Those outer circles are tangent to the incircle and respective sides. I have tried and failed ...
0
votes
1answer
87 views

Transformation of 2D profile to 3D coordinates

I am sure that answer for similar questions have being given more than one thousandth times, but correct answer that suits my needs I still haven't found. Currently I am developing simple 3D app. My ...
1
vote
2answers
37 views

Looking at an angle rotated

Suppose you have an angle of degree theta painted on the ground at a spot. You are standing d distance away and looking at it from a height of h and from your perspective the angle appears to be of ...
0
votes
1answer
976 views

convex hull function in matlab

Is there any way to compute the convex hull of a finite set of points in Matlab and gives the half-space representation as its result? I usually use a toolbox called MPT developed at ETHZ Zurich, but ...
0
votes
1answer
79 views

Altitudes of Triangle

I have a triangle defined as 3 lines, each defined by two coordinate points A and B. I have the area of the triangle but need to calculate the 3 altitudes and their respective sides A and B points. ...
5
votes
1answer
278 views

How can a simple closed curve not look locally like the rotated graph of a continuous function?

A simple closed curve is a continuous closed curve without self-intersections. The question of whether you can inscribe a square in every simple closed curve is currently an open problem, but this ...
0
votes
2answers
4k 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 ...
2
votes
3answers
49 views

Trigonometric problem: Elevation angle [closed]

The elevation of the top of a tower $KT$ from a point $A$ is $27^\circ$. At another point $B$, $50$ meters nearer to the foot of the tower where $ABK$ is a straight line, the angle of elevation is ...
0
votes
0answers
98 views

compact surface of revolution

I have to prove that a compact surface of revolution is diffeomorphic to a sphere or to a torus. And show that $\int_{S} K dA= $ =$\{4\pi,$ if S is spherical type 0, if S is toric type $\}$, ...
4
votes
2answers
135 views

How find the $AP+\frac{1}{2}BP$ minmum value

An equilateral triangle $ABC$ such $$AB=BC=AC=2a>0$$ A circle $O$ is inscribed in triangle $ABC$,and the point $P$ on the circle $O$. Find the minimum $$AP+\dfrac{1}{2}BP$$ My idea: let ...
0
votes
1answer
69 views

How many tetrahedrons an edge belongs to in a Body-Centered Cube Lattice?

Body-Centered Cube Lattice, also known as BCC Lattice (shown in the Figure) is a kind of lattice where the body of a cube also contains a vertex. 4 vertices as the figure shows constitute a ...
2
votes
0answers
117 views

Proving there is no set of five distinct points s.t. every three points are the vertices of a right triangle.

We can see that the following proposition is true. Proposition : Each triangle $ABD, ACD, BCD$ is a right triangle for $$A(0,b,0), B(a,0,0), C(0,0,0)\ \ \ (a\gt 0, b\gt 0)$$ $\iff D$ is either ...
1
vote
2answers
3k views

Find perimeter of traiangle when you know two sides

Here's the GRE problem: The length of one side of a triangle is 12. The length of another side is 18. Which of the following could be the perimeter of the triangle? ...
1
vote
2answers
114 views

Flat Surfaces in $\mathbb{R}^3$ Can Be Bent Only Along Straight Lines

This is a problem out of Elementary Differential Geometry by Barrett O'Neill (Chapter 6 Section 3 Number 2). Let $M$ be a flat surface in $\mathbb{R}^3$ with principal curvatures $k_1$ and $k_2$, ...
1
vote
3answers
430 views

Show that there are exactly two lines through a point p outside the circle that are tangent to the circle C

Let $C$ be a circle of radius $r$ in the plane. Let $p$ be a point in the plane that lies outside of $C$. Show that there are exactly two lines through $p$ that are tangent to $C$. It is one of ...
0
votes
3answers
562 views

Is there any geometry where ratio of circle's circumference to its diameter is rational?

In Euclidean geometry, the ratio of the circumference of a circle to its diameter is an irrational number, 3.14159 and so on. But if you change to non-Euclidean geometries, you get other values for ...
1
vote
0answers
24 views

Integrating along a line of point sources

I have some concentration that radiates from a spherical point, being steadily consumed until it hits zero at some distance $r_{n}$. This is given by $$C(r) = A\left[r^2 + \frac{2r_{n}^3}{r} - ...
1
vote
1answer
58 views

Is it true there exists $f:S^{2n}\longrightarrow S^{2n}$ making the diagram commutative?

Let $g:\mathbb R\mathbb P^{2n}\longrightarrow \mathbb R\mathbb P^{2n}$ be a continuous map where $\mathbb R\mathbb P^{2n}=\mathbb S^{2n}/\{\pm x\}$. Is it true there exists $f:\mathbb ...
1
vote
2answers
204 views

Maximize the distance between a point and a bounding rectangle

There are $n$ random points in the $x-y$ plane, whose coordinates are known beforehand. We can use a minimum bounding rectangle (MBR) to bound these points. In this scenario, the MBR can be rotated, ...
1
vote
1answer
69 views

The simplest way to calculate area of a pentagon

I have a pentagon, whose all sides and angles I know. What would be the simplest way, i.e requires least calculations, to calculate its area? If possible, can I generalize your way to higher ...
0
votes
1answer
29 views

$C^{-1} (1+|x|^{2})^{\frac{s}{2}} \leq (1+|x|)^{\frac{s}{2}} \leq C (1+|x|^{2})^{\frac{s}{2}}$?

Let $s\in \mathbb R,$ and define $f: \mathbb R^{n}\to [0, \infty)$ such that $f(x)= (1+|x|^{2})^{\frac{s}{2}}, (x\in \mathbb R^{n})$ and $g:\mathbb R^{n}\to [0, \infty)$ such that $g(x)= ...
0
votes
1answer
59 views

Tangent of a curve

Consider the curve $x=1$ in $xy$ plane. I want to know whether tangent at any point on this curve exist which is $x=1$, or tangent does not exist.
1
vote
0answers
30 views

To construct a right triangle given the hypotenuse and sum of two legs [duplicate]

NOTE: I want a hint only. A compass and a straightedge construction:Given a hypotenuse and the sum of lengths of the legs,we need to construct a right triangle. MY TRY: From any ray $BE$, ,let ...
3
votes
0answers
294 views

How to find the minimal path between points in a planar set with holes in it?

When I was a commuter student, I would park in a very large parking that that had a set of stairs in a corner that I had to climb. In general, I had to park far away from this corner in an almost full ...
2
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1answer
140 views

Algebra, Geometry and Algebraic Geometry

I want to know, what is the difference between Algebra, Geometry and Algebraic Geometry ? Your reply is highly appreciated.
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3answers
237 views

How many square based pyramids are in a bigger pyramids?

The biggest challenge to solve the problem is that I can't really picture a pyramid. And it is hard to make a model. The pyramids I am trying to find include those on all tiers.
0
votes
1answer
89 views

How to find the equation of a line which intersects these lines at 90 degrees?

How to find the equation of a line which intersects these lines at 90 degrees? $p\equiv \dfrac{x}{2}=\dfrac{y+1}{0}=\dfrac{z-2}{1}$ $q\equiv \dfrac{x-1}{1}=\dfrac{y-2}{1}=\dfrac{z+5}{0}$ Since the ...
2
votes
1answer
105 views

Finding distance of point from 4D ray

I'm working on a programming project. In this project, a ray is fired from a point in 4-space. I need to find the distance from this ray to a number of other points in 4-space. I attempted to solve ...
0
votes
1answer
100 views

How to find how many rectangular prisms ( including cubes) are in a n by n by n cube?

I somehow got the answer to be [(n+1)!/2!(n+1-2)!]^2 *n Each part of the equation represents the height, length, and width of the possible rectangular prism in the big cube. You can multiply the ...
1
vote
1answer
54 views

Order of $A/2A$ for $A$ an Abelian variety

Let $A$ be an Abelian variety over $\mathbb R$ of dimension $g$. Then the size of $A(\mathbb R)/2A(\mathbb R)$ is $(\# A(\mathbb R)[2])/2^g$. I'm wondering how one might go about proving such a ...
0
votes
1answer
42 views

Finding angles in Barycentric system

How to find the angles of a triangle given the barycentric coordinates of its corners? Does it work if i take the first two components of every coordinate, and find the angles in the triangle (on the ...