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
0answers
15 views

An algorithm for filling a moving truck

I was recently helping a friend move. I stood in the moving truck as other people brought boxes and furniture pieces from inside the house. My job was to arrange these items in an efficient way inside ...
9
votes
2answers
58 views

Finding $\lim_{x\to 0} \frac {2\sin x-\sin 2x}{x-\sin x}$ geometrically

While looking at this question, I noticed an interesting geometric interpretation of the limit the OP was trying to evaluate. His limit came to twice the value of the limit $$\lim_{x\to 0}\frac{\sin ...
1
vote
1answer
31 views

Geometry formulas, how to show identities.

Given $d$ is integer: How do I show: $$\frac{1}{(e^{\frac{2i\pi p}{d}}-1)}=\frac{-i}{2\tan(\frac{\pi p}{d})}-\frac{1}{2}$$ How do I rewrite and show, for $k$ is an integer: $$ ...
-1
votes
0answers
19 views

perpendicular prove with similarities in triangles

$△ ABC$ is given . In line through the vertex A and perpendicular to side BC two points $A_1$ and $A_2$ are taken so that $AA_2 = AA_1 = BC$ ($A_1$ is closer the line $BC$ than $A_2$ ). In the same ...
0
votes
1answer
26 views

Given only angles and area of triangle, find side length.

The area of a triangle is $60$ square inches. Find the length of the side included between $A = 25°$ and $C = 110°$. (Round your answer to one decimal place.)
-4
votes
3answers
59 views

Prove that $\sin^2 \theta + \sin^2 \beta= \sin(\theta + \beta)$ when $\theta+\beta = 90^\circ$

If $\theta, \beta$ are two acute angles prove that : $$\sin^2 \theta + \sin^2 \beta= \sin(\theta + \beta) $$ when $\theta, \beta$ are complementary angles, i.e. $\theta + \beta = 90°$. My try... ...
2
votes
1answer
10 views

Volume of the symmetric difference between a parallelotope and its translated.

Let $A$ be a n-dimensional parallelotope and $v \in \mathbb{R}^n$ a vector. Is there a formula giving the volume of the symmetric difference $A \Delta (v+A)$?
2
votes
3answers
99 views

Find a point so that the triangle is equilateral

We have O(0,0), A(3,4) and B(x,y). Find $x,y\in{R}$ so that the OAB triangle is equilateral. I tried using the fact that the median is also the altitude(height) of the equilateral triangle. I ...
0
votes
1answer
22 views

Peripendicular Line at distance d from point in a given direction

I have a line given by $Ax + By + C= 0$, and a point $x_0,y_0$. From that point $x_0,y_0$ in the direction of the line up to distance $d$, I want to find the equation of the line that is perpendicular ...
0
votes
0answers
13 views

How can I measure segment of a shapes perimeter?

I am creating a tracing tool and one aspect of that is that I need to determine the length of a segment of a shape's outline. Given any shape within a 100x100 grid, and two points on that shapes ...
2
votes
0answers
20 views

spherical segment volume

Suppose I have a spherical segment like the one in the picture. I want to find the infinitesimal volume of such a segment. The angle between point A and B is $d\theta$. And the radius of the sphere ...
0
votes
0answers
18 views

Sharper Bound for Minkowski's Convex Body Theorem

To satisfy the conditions of Minkowski's Convex Body Theorem we need a lattice $\Lambda$ with fundamental domain $T$ and a bounded convex symmetric subset of $\mathbb{R}^n$ (call it $X$) with ...
0
votes
4answers
28 views

Points $A$, $B$, and $C$ are on the circumference of a circle with radius 2

Points $A$, $B$, and $C$ are on the circumference of a circle with radius $2$ such that $\angle BAC = 45^\circ$ and $\angle ACB = 60^\circ$. Find the area of $\triangle ABC$. I've drawn a circle ...
1
vote
4answers
46 views

Probability question involving infinite number of vertical chords in a 1 inch circle.

Infinite number of vertical chords drawn on a circle with a 1 inch radius. What is the probability that a randomly picked chord is shorter than the radius? The answer should be $1 - .5√ 3$ or ...
0
votes
1answer
42 views

An isosceles right triangle has legs of length 10. A pin is dropped into it and lands somewhere in the triangle where all places are equally likely.

What is the probability that it does not land within 2 units of any of the sides? From my calculations, I get that the smaller triangle has side lengths of 4,4, 4 root 2 (-2 at the right angle and ...
-9
votes
2answers
86 views

Ayn Rand and athematics

I am an honors undergraduate in mathematics. I have taken an interest in objectivism. I came across a discovery of Ms. Ayn Rand's in mathematics: In a triangle the inscribed circle touches the ...
1
vote
1answer
16 views

Finding grid nodes a line passes through

For 2D grid pathfinding, I want to do a quick broadphase to check if there is a direct path from the start to the target by conceptually checking all nodes touching the line segment formed by ...
13
votes
0answers
83 views

Is Tolkien's Middle Earth flat?

In the first introductory chapter of his book Gravitation and cosmology: principles and applications of the general theory of relativity Steven Weinberg discusses the origin of non-euclidean ...
12
votes
3answers
142 views

Product of cosines: $ \prod_{r=1}^{7} \cos \frac{r\pi}{15} $

Evaluate $$ \prod_{r=1}^{7} \cos {\dfrac{r\pi}{15}} $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\cos(A-B)] ...
2
votes
2answers
24 views

Local existence of parallel vector field

Let $M$ be a Riemannian manifold, $p\in M$ be a point in the manifold, and $\xi\in T_p M$ be a vector in its tangent space. I am wondering whether, for some small neighborhood $U\subset M$, it is ...
0
votes
0answers
14 views

How to trace the path of a moving point in geogebra

I created a simple animation in geogebra, two intersecting lines rotating around fixed points. I want to trace the paths of the vertex points of these lines. I mean is there a way when the animation ...
0
votes
2answers
18 views

Algorithm for intersection of n circles with approximate values

I'm trying to come up with a sort of trilateration algorithm that, given n >= 3 circles, finds the point of intersection. The radii come from samplings of electromagnetic magnitudes, therefore there ...
5
votes
3answers
72 views

Circles revolving around each other and infinities

I just watched this video, and I'm a bit perplexed. Problem: ...
3
votes
2answers
78 views

How to solve an equation involving euclidean norm operation?

On page 3 of Scalable, Versatile and Simple Constrained Graph Layout it describes the equation: $$|(\mathbf p-\mathbf r)-(\mathbf q+\mathbf r)|=d$$ Where $\mathbf p$ and $\mathbf q$ are known ...
-1
votes
1answer
16 views

Area of a quarter circle C1 equals the area of an inner circle C2 where C2.diameter = C1.radius

Say we have a circle C1 with radius 2. Inside of that we draw circle C2 going from the centre point of C1 to the perimeter of C1 (making it diameter = 2) ...
0
votes
1answer
33 views

compact image of a continuous function from compact set to C

Suppose that we have a continuous function $h:[0,1] \times [a,b] \to G$, where $G$ is an open subset of $\mathbb C$. Prove that we can partition $[0,1]$ and $[a,b]$ to $\{x_0, x_1, \ldots, x_n\}$ ...
2
votes
1answer
33 views

Diameter of a 10-ball in a 10-box is larger than the side length of box?

I came across this idea in a lecture on elementary topology. While it makes sense algebraically, I'm hoping someone could shed some light on the way this is possible. So you begin with a square of ...
1
vote
1answer
19 views

What's the name given to the ratio $P^2/A$ for a closed figure in the Euclidean plane?

Let $\mathscr{F}$ be the set of all plane, closed Euclidean figures having positive perimeter, and let $\sim$ be the similarity relation on $\mathscr{F}$. Then, for any equivalence class ...
3
votes
1answer
53 views

Area of the shaded part in rectangle

The question asks you to determine the shaded part of the rectangle in terms of x. please will someone help with this problem, i have spent a while on it with not much progress.
1
vote
1answer
23 views

Get the Equation of a Plane from a Vertex and 2 Angles?

What is the simplest way to algebraically get the equation of a Plane (ax + by + cz = d), if you only have 1 point on the plane, and 2 angles (horizontal and vertical) which define the direction the ...
2
votes
5answers
125 views

Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$”

If $C$ and $A$ are the circumference and area of a circle, then $$ \frac{C^2}{A}=4 \pi\; . $$ I'm looking for a reference to an elementary but rigorous proof of it. I'm particularly interested in ...
4
votes
2answers
83 views

Show that in any triangle, we have $\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$

Show that in any triangle, we have $$\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$$ where $R$ is the circumradius of the triangle. Here is my work: ...
0
votes
1answer
26 views

Transformation of axes by rotation

How can I intutively understand the formula for getting new coordinate of point P after rotation of axes which was P(x,y) with respect to the old axes?
1
vote
1answer
19 views

Arcs and surfaces. Why are there finitely many arcs on the surface up to the action of MCG?

Given a bordered surface $S$ (I imagine this is true for non-orientable surfaces too, but you may restrict to the case of orientable surfaces) with finitely many marked points on each boundary ...
-1
votes
1answer
30 views

Parabola problem [on hold]

Water squirting out of a horizontal nozzle held $4$ ft above the ground describes a parabolic curve with the vertex at the nozzle. If the stream of water drops $1$ ft in the first $10$ ft of ...
1
vote
0answers
22 views

About Homothetic transformation

I have one question regarding the way which a homothetic transformatior is written. Why is it written in the following way: $$\vec{OH^{k}_{O}(P)}=k\vec{OP}+(1-k)\vec{OO}?$$ From where ...
1
vote
1answer
23 views

logarithmic spiral around cone stump

Based on the answer on my previous question I managed to come up with the following equations: $$\begin{eqnarray} k &=& 1 \\ r_\Delta &=& r_b - r_t \\ r(\theta) &=& r_t * ...
0
votes
0answers
19 views

How does the steepness of lines through a hyperboloid change the further away they are from the apex?

I'm a geoscientist and am trying to figure out how the steepness of the flanks of a hyperboloid change for straight lines that cross them. The line of reference is through the apex. Basically any ...
-5
votes
0answers
44 views

If I invent(in maths) something , Then I want to give it to world, what is process? [on hold]

If I invent(in maths) something , Then I want to give it to world, what is process?
0
votes
0answers
13 views

Determine when an object moving along a line crosses a constantly rotating ray

I'm trying to make a visual experiment akin to a radar, where there are 'ships' moving across lines in a $2$-dimensional space and in the center of that space is a ray extending outward, which ...
1
vote
1answer
29 views

How high above sea level do your eyes have to be to see a point that is 4.1 miles away “as the crow flies”?

There's a fireworks show going on tonight at a little town that's 4.1 miles away from my house, and I want to watch it from a hill near my house. So I thought I'd set up a simple geometry problem to ...
6
votes
1answer
53 views

$\frac{MA}{BC}+\frac{MB}{CA}+\frac{MC}{AB}\geq \sqrt{3}$

Given ∆$ABC$ and $M$ is an interior point.Prove that: $\dfrac{MA}{BC}+\dfrac{MB}{CA}+\dfrac{MC}{AB}\geq \sqrt{3}$ When does equality holds?
2
votes
1answer
39 views

Collision between moving circular discs

I am trying to figure out how to detect collision between two moving circular discs that move along a pretedermined path with a known speed. Example: Circular disk $A$ with radius $r1$ moves along ...
-1
votes
2answers
63 views

Six variables. System of equations.

$$ \begin{align} x & =\frac{R+\frac{G+B}{-2}}{R+G+B} \\[10pt] y & =\frac{\frac{(G-B) \sqrt{3}}{2}}{R+G+B} \\[10pt] z & =R+G+B \end{align} $$ How do I get the formula for ...
2
votes
1answer
56 views

inscribed circle in $n$-gon

If I'm given a circle with radius $r$ and I want to create a polygon with side $n$ (say $n=5$) which can cover the circle fully, then how to prove that a regular polygon is the solution with minimum ...
0
votes
0answers
28 views

Relation between farthest pair of points and closest pair of points in plane

I am writing program for obtaining distance between shortest and farthest pair of points among the given points in plane .I am able to calculate them both the shortest one using divide and conquer ...
2
votes
1answer
17 views

Find intersection of non-parallell planes without further assumption on their normals

Finding the intersection line between two planes is basic linear algebra but is it possible to find one formula, without having to dealing with different cases? Example: $$ \left\{ \begin{aligned} ...
1
vote
1answer
22 views

Checking whether points form a polygon in complex plane

If z^8=(z-1)^8 then the roots are 1) concyclic 2) form a polygonal 3)none I found the roots to be 1+cot(k.pi/8) for k is a natural number and less than 8. Then couldn't figure it out.
1
vote
0answers
227 views

Find radius of Circle

There is a circle C1 of Radius R1 and another circle C2 of radius R2 (R2 ≤ R1) such that it touches circle C1 internally There is another circle C3 with radius R3 such that it touches the circle C1 ...
1
vote
1answer
49 views

Area of overlapping squares

I'm working on a programming project and got to the point where I need to find how much is the blue square overlapping each of the other 9 squares. The squares' sides(including the blue one's) are ...