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
20 views

Point in a rectangle

$ABCD$ is a rectangle and $P$ is a point in the same plane. If the perpendicular through $C$ to $AP$ and the perpendicular through $B$ to $DP$ intersect at $Q$, prove that $PQ \parallel AD$. ...
0
votes
1answer
65 views

Finding the value of k

If $x,y,z$ are perpendicular distances from circumcenter on the sides $BC,AC$ and $AB$ respectively. In need find $k$ such that $$\frac ax+\frac by+\frac cz=\frac{abc}{kxyz}$$ (Lowercase letters ...
2
votes
1answer
39 views

Finding the third side of a triangle, given ratio of two sides and difference of two angles [closed]

Given $a=2b$ and $|\angle A-\angle B|=60$ degrees. Find the third side, where lowercase letters denote opposite sides and uppercase letter angles. Progress I could find the $\cos C$ but then ...
2
votes
1answer
34 views

Properties of the simplest object in n-dimension

In my boredom, I was thinking about why the simplest 3d object (i.e. the one with the least faces, sides, vertices) was the tetrahedron. After it made sense to me, I realized some cool stuff which was ...
0
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1answer
20 views

hanging pictures: a practical question about horizontal centering on a wall

Here's a little math/physics problem I just ran into with some house maintenance. Suppose you want to hang a heavy picture/mirror in the center of a wall. However, the studs are not arranged in a way ...
0
votes
1answer
25 views

Find the equation of parabola tangent to a line

I know how to find the equation of the line tangent to a parabola through a certain point. But how do I find the equation of the parabola from the point and the tangent line? For example, how do I ...
4
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0answers
60 views
+100

Law of sines: uniform proof of Euclidean, spherical & hyperbolic cases

There is a unified formulation of law of sines which is true in all 3 constant curvature geometries (Euclidean, spherical, hyperbolic): $$ \frac{l(a)}{\sin\alpha}= \frac{l(b)}{\sin\beta}= ...
0
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1answer
19 views

On Euclid's definition of similar and equal solid figures.

The Euclid's definition of similar solid figures is Similar solid figures are those contained by similar planes equal in multitude. And the Euclid's definition of equal solid figures is Equal and ...
0
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2answers
38 views

Question about sines of angles in an acute triangle

Let ABC be a triangle such that each angle is less than 90 degrees. I want to prove that sinA + sinB + sinC > 2. Here is what I have done: Since A+B+C=180 and 0 < A,B,C < 90, at least two of ...
0
votes
0answers
17 views

Distance between point and translated subspace

Given $(m - 1)$-dimensional subspace of $n$-dimensional space ($m \leq n$), that is defined by a set of $m$ its (linearly independent) points. How to compute the distance between separate point $p_0$ ...
1
vote
2answers
44 views

Geometric Construction Problem

Given three non-collinear points A, B, and C, construct three circles that are pairwise tangent at these points. Are there any cases where such circles do not exist? I am not sure how to start the ...
5
votes
3answers
61 views

Proof that $\cos(\pi/4)=\frac{\sqrt2}{2}$

Normally I just look up or remember that $\cos(\pi/4)=\frac{\sqrt2}{2}$, or type "$\cos(\pi/4)$" into WolframAlpha to check the answer. But what about the first time someone wanted to know what ...
2
votes
3answers
43 views

Triangle of maximum perimeter for a given area

What type of triangle has the maximum perimeter for a certain area? Suppose I start with a rectangle of that area (axb=Z). I can stretch one dimension of the rectangle until infinity, reducing the ...
0
votes
1answer
18 views

How to find horizontal shift after solving catenary

I'm trying to compute the chain/rope/string curve between two points and a given length. I followed the instructions as answered here but I have a last step to accomplish. The equations I'm solving ...
1
vote
0answers
25 views

Showing that vector field $\mathbb{Z}$ satisfies $\mathbb{Z}\cdot(\nabla \times \mathbb{Z})=0$,connected to the Frobenius Theorem

Suppose $\mathbb{Z}$ is a smooth vector field on $\mathbb{R}^3$ with $\mathbb{Z}^3(x,y,z) \neq 0$. a) Find functions $f$ and $g$ such that the vector fields $\mathbb{X}=(1,0,-f)$ and ...
1
vote
0answers
34 views

Does this equations represent the sphere?

Consider complex numbers $x_1$ and $x_2$. Let $j=\sqrt{-1}$. Consider the set of points $$\mathcal{S}=(x_1x_2^*+x_1^*x_2,j(x_1x_2^*-x_1^*x_2,|x_1|^2-|x_2|^2)$$ for each point $(x_1,x_2)$ such that ...
0
votes
0answers
24 views

How to find unknown coordinates form known coordinates and distance

So i am given this question where the start and end point coordinates of a line are known say; 34.44,71.50 and 35.66,71.35, and i am asked to find unknown coordinates at 100ft distance from either ...
6
votes
3answers
253 views

Trisecting an angle

In this Numberphile video it is stated that trisecting an angle is not possible with only a compass and a straight edge. Here's a way I came up with: Let the top line be A and bottom line be B, ...
2
votes
1answer
57 views

Geometry Problem about tangent lines

Let S be the circumcenter of ABC. $A_0$ is the middle of arc BC not containing A, $C_0$ also the middle of arc AB without C. Let $S_1$ be a circle with center $A_0$, tangent to BC, $S_2$ with center ...
-1
votes
0answers
93 views

Tessellation of a grid with squares and triangles

A set S of unit squares is chosen out of a large grid of unit squares. The squares of S are tiled with isosceles right triangles of hypotenuse 2 so that the triangles do not overlap each other, do not ...
0
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0answers
150 views

Numbers put around a circle

Nine distinct positive integers are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of n and the product of any two adjacent numbers in the ...
1
vote
1answer
47 views

Probability of being the sides of quadrilateral

Divide a given line segment into two other line segment. Then cut each of these new line segment into two more line segment. What is the the probability that the resulting line segment are the sides ...
2
votes
1answer
39 views

Equivalence of geometric and algebraic definitions of conic sections

I have not been able to find a proof that the following definitions are equivalent anywhere, thought maybe someone could give me an idea: A parabola is defined geometrically as the intersection of a ...
2
votes
1answer
17 views

points in general position

I'm really confused by definition of general position at wikipedia. I understand that the set of points/vectors in R^d is in general position iff every (d+1) points are not in any possible hyperplane ...
1
vote
1answer
55 views

How can I determine the center of this circumference?

I have the following question: if I have an irregular symmetric polygon, how can I determinate the circumference with the least area that contains this polygon? I believe (in case that the polygon ...
0
votes
1answer
58 views

Given two points and two normals, how to find third point

I really don't know how to search for this specific question. So, I'll try my best to explain my issue. I have the point P1 (pink) and the normal vector M (white) of its line, Given an ...
7
votes
3answers
111 views

prove that quadrilateral is cyclic

let $ABCD$ be a rectangle with $BC=2AB$.Let E be the midpoint of side BC and P an arbitrary inner point of AD. Let F and G be the feet of perpendiculars drawn from A to BP and from D to CP. Prove that ...
0
votes
2answers
48 views

Hexagonal Tessellation on a sphere

I want to detect collision of a sphere with another object and to find out(show) the deformation of the sphere. I have come to know that hexagon(regular)tessellation of a sphere is the most ...
0
votes
2answers
39 views

Rotation to obtain corodinate

Given: $x0$ , $y0$, $x1$ and $y1$ points and angle $A$. How to compute $x2$ in the figure specified. Sorry if the questions seems trivial. I have forgot much of geometry.
2
votes
1answer
33 views

Angles of diagonals in a quadrilateral

I have a quadrilateral with known angles. Also known is that edge BC and CD have the same length. How can I find out the ratio the diagonals divides the angle α into α1 and α2?
0
votes
1answer
26 views

Rotate a line by a given angle about a point

Given the coefficients of a line $A$ , $B$ and $C$. $$ Ax + By + C = 0$$ I wish to rotate the line by angle say $\theta$ about a point $x_0$ and $y_0$ in clockwise direction. How can I achieve this ...
3
votes
2answers
26 views

symmetry group of hypercube in $\mathbb{R}^4$ [closed]

Let $ = \{(x, y, z, w) \in \mathbb{R}^4 \text{ }|\text{ }|x|,\, |y|,\,|z|,\,|w| \le 1\}$ be the hypercube in $\mathbb{R}^4$ of side length $2$ centered at the origin. Identify the symmetry group of ...
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vote
0answers
25 views

Find 3D concave hull based on original model and convex hull

I want to find the concave hull of a 3d model, with a threshold for the maximum edge size. Googling around let me to the following approach (mainly abstracting from 2d approaches): Determine the ...
0
votes
1answer
29 views

Prove that A + B is invertible iff $I_n$ + $A^{-1}$B is invertible (matrices)

We are given that a matrix $A$ in $R^{n\times n}$ is invertible. We must show that $A + B$ (also in $R^{n\times n}$) is invertible if and only if $I_n$ + $A^{-1}$$B$ is invertible. I cannot figure ...
0
votes
4answers
27 views

Finding the equation of a circle given one point and radius

What is the equation of the circle which passes through the point $(0,2)$ with radius $4$ and whose center lies on the line $y = x$?
0
votes
1answer
29 views

What is the radius of a circle tangent to two lines with a known angle between them

Given angle, $\alpha$, and distance, $d$, what is the radius, $r$, and angle, $\theta$, in the image below in terms of the known quantities?
0
votes
1answer
20 views

Show the Boundary of a translation is the translation of the Boundary

Define the translation of a set $E$ as: $$a+E=\{z\in \Bbb R^2: z=a+x,\text{ for }x\in E \} $$ I need to show that $\delta (a+E)= a+ \delta E$, where $\delta$ denotes the boundary of the set. i.e ...
1
vote
1answer
21 views

I think its Rotational Kinematics

I have a post two meters high with a diameter of ten (10) centermeters. I rope is wrapped completely around the post (bottom to top). The rope has a diameter of two (2) centermeters. A crow picks ...
0
votes
1answer
30 views

Euclidean isometries

I am asked to show that every translation of the euclidean plane can be written as two reflections. How do I proceed (algebraicly)? My idea is to proof it in a sense of creating a rectangular ...
4
votes
5answers
73 views

How to geometrically prove the focal property of ellipse?

How to prove geometrically that if we have a tangent of ellipse with focus F and F' in point P, that tangent is bisector of the angle between a line joining focus F to point P and the line F'P outside ...
1
vote
2answers
10 views

Getting the ratio that an oriented segment is divided by a line

I am hardly trying to understand vectors, but it is even harder than I expected... I have the following drawing: We have: $ABCD - square$ $M $ - middle of $(CD)$ $DMNP - square$ Determine: ...
3
votes
1answer
30 views

Fubini type results for Hausdorf dimension?

Suppose that I have a stack of hyperplanes in Euclidean space $\mathbb{R}^n$, let's call each plane $P_{a, x}=\{y\in\mathbb{R}^n\mid \langle y, x\rangle=a\}$ Suppose that a measurable subset $A$ of ...
1
vote
0answers
42 views

How to find mass points and ratios in a triangle?

How to find mass points with weights and ratios is my question. In my class, we learned about mass points. First we had the given ratios of 2 side lengths. Given: MC = d MB = e MA = f BD:DA = ...
1
vote
5answers
116 views

Why sin of supplementary angles have equal values?

Given two supplementary angles (for instance, 30 degrees and 150 degrees), why is $\sin(30^\circ) = \sin(150^\circ)$? Where can I find a proof for this? Or the derivative of such proofs?
2
votes
2answers
31 views

Finding linear transformation such that $\operatorname{im} \phi = \ker \phi = \operatorname{span}(\alpha_1, \alpha_2)$

Here i am completely lost. I have to find a formula for linear transformation $\phi : \mathbb{R}^4 \rightarrow \mathbb{R}^4$ such that $\operatorname{im} \phi=\ker \phi = \operatorname{span} ...
0
votes
3answers
68 views

What's the best polygon for tiling the plane?

We want to cover the whole plane by tiles, shaped as a polygon with equal-length sides, such that there is not overlapping and any gap (Note that all the tiles are similar to each other). which ...
1
vote
1answer
28 views

An altitude is divided into 5 equal parts by 4 lines. Prove that the the areas of alternate sections are equal.

The question is as follows : Let their be a triangle ABC. Make altitude AD on C. Divide this altitude in 5 equal parts with lines EF, GM, IJ, KL intersecting at points M,N,O,P respectively. We have ...
3
votes
1answer
71 views

What exactly is the 'tension' between arithmetic and geometry?

We all know Pythagorean theorem, $a^2+b^2 = c^2 $ Im reading John Stillwell, Mathematics and its history at the moment, and during the greek antiquity they had some trouble by interpretating ...
3
votes
3answers
64 views

Find equation of a circle

Find equation of a circle passing through $(1, 1)$ and touching the circle $$ x^2 + y^2 + 4x - 6y - 3=0 $$ at the point $(2, 3)$. I am stuck as I cannot find more than $2$ equations for $3$ unknowns. ...
0
votes
1answer
40 views

Mass of Solid, Multivariable Calculus

Calculate the mass of the solid that lies above the surface $z= 0$, below the surface $z=y$, and inside the surface $x^2+y^2 = 4$ with the given density $yz$. I have switched to cylindrical ...