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

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Geometry Transformation Problem

Given two circles C1 and C2, line s, point D, and point E like the image below : Explain me the procedure to find point P on C1 and Q on C2 such that the length of PQ = the length of DE and line ...
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9 views

In Neutral Geometry, prove that the opposite sides of a rectangle are congruent.

I'm having some trouble proving a theorem of Neutral Geometry. First, allow me to clearly state what we are allowed to assume in Neutral Geometry: Hilbert's incidence axioms Hilbert's order axioms ...
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8 views

txml question help

In the coordinate plane, the vertices of a quadrilateral are (-4,-2), (8,5), (6,8), and (-1,6). What are the coordinates of the point in the plane for which the sum of the distances from this point to ...
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1answer
39 views

Easiest way to verify that $4x^2+y^2=1$ is an ellipse?

Normally I would just divide both sides by the number $4$ because it's not good in there, but I can't do it for $$4x^2+y^2=1$$ I must have $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ So what's the ...
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1answer
25 views

Diameter of the Grassmannian

Just an interesting question that came to my mind while studying(!): Since the Grassmannian $G(k,\mathbb{C}^n)$ is a compact manifold, what do we know about its diameter? Do we know any estimate? ...
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9 views

Weighted mean of an object (Centers of mass)

I am having trouble understanding the concept. Usually when I calculate the center of mass of an object when given area and dimensions I'd multiply corresponding distances with areas etc then ...
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1answer
38 views

Some question about path connectedness

I think that's intuitively evident but I can't prove that the set $\mathbb{S}^n\setminus\{(0,\cdots, 1),(0,\cdots, -1)\}\; (n>1)$ is path connected. Does anyone have a formal argument to prove it? ...
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1answer
16 views

Length of curves with same images

From a geometrically intuitive point of view, it is obvious that if two injective $C^1$ curves $\gamma,\delta$ with values in $\mathbb R^n$ have the same images, then their lengths $\ell(\gamma)$ and ...
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24 views

find the corect angle to cut pipe

I have pipe penitrating a wall at an angle shooting up. I have to attach a 90 degree ellbow an drop from 90 plumb,so I will have to cut the unlevel pipe on an angle to achive a plumb drop from 90. How ...
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16 views

Using azimuthal and polar angles in ECEF coordinate system

I have a physical cone, which its vertex located in some point (x,y,z) in ECEF coordinates, and I want to check if another point is inside this cone. In order to do it, I have to take into ...
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1answer
38 views

Circle from three points on surface of sphere

I need to compute the circle on the surface of a sphere given three points on that very surface. It is very easy to do that in Euclidean geometry, but the sphere has no x and y, but just two angles ...
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14 views

Area of a bounded region $R$ of surface $z = f(x,y)$ is $A = \int\int_{Q} \sqrt{1 + (\partial_x f)^2 + (\partial_y f)^2}\; dxdy$

This is Exercise 5 from do Carmo section 2.5 (page 100). Area of a bounded region $R$ of surface $\left\{ z = f(x,y)\right\}$ is $A = \int\int_{Q} \sqrt{1 + (\partial_x f)^2 + (\partial_y f)^2}\; ...
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23 views

Repeatedly interpreting polar coordinates as Cartesian

Start with a Cartesian point $(x,y) \in \mathbb{R}^2$, convert it to polar coordinates $(r,\phi)$ ($\phi$ in radians), and then reinterpret $(r,\phi)$ as $(x,y)$, i.e., set $$(x,y) = (r,\phi) \;.$$ ...
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1answer
46 views

confused about how adding inches makes it square [on hold]

I have a fountain base that was 36 inches wide and 16 inches deep. I added 4 inches to both making it 40 inches wide and 20 inches deep. I am confused as to before at 36 and 16 it was not perfectly ...
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3answers
31 views

Volume of a Rectangular Prism when given only the Surface Area

So, in my equation, they ask you to find the volume of a rectangular prism when you only are given the surface area. As an example, one of the equations gave you a surface area of 240 yards squared, ...
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3answers
48 views

Can the formula for finding the slope of a line be reversed and still be right?

The formula for finding the slope of a line is $y_2-y_1\over x_2-x_1$, but can it be reversed into $y_1-y_2\over x_1-x_2$ and still be right?
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26 views

Complex singularity exponent

I am studying about complex singularity exponents of holomorphic functions. I need some help to clarify a few things: First, what a complex singularity exponent is, for the holomorphic function ...
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3answers
19 views

Calculate sides of right triangle with hypotenuse and area or perimeter

I'm trying to find if it is possible to find the lengths of the base and height of a right triangle with only the hypotenuse and the area (or the perimeter) of the triangle. I would have just figured ...
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2answers
41 views

Can some one explain why the answer to part a describes a circle, or part of it?

Problem Statement: The transformation $T$ from the complex $z$-plane to the complex $w$-plane is given by $w=\frac{z+1}{z+i}, z\neq i$. a) Show that $T$ maps points on the half-line $\arg ...
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1answer
23 views

Sketch $\text{Arg}( 1 - z ) = \pi/2$ and find the Cartesian equation

This is a question I was thinking about when doing Edexcel FP2, seen something like it before but not sure where. Tried to solve it but cannot.
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1answer
20 views

How do i scale 2D vector using matrix

I know that scale matrix is 2x2 { x, 0, 0, y } basis. My vector { 100, 2 } and i want to scale it using custom 2x2 matrix. I've read that if left operand is 2D row vector, then multiplying it on a ...
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3answers
55 views

Construct the great circle (geodesic) in spherical or Riemanian geometry

Given: a circle $C$ with centre $M$ two points $P_1$ and $P_2$ inside circle $C$, so that $M$ is not on the line $P_1P_2$. Cunstruct an other circle $O$ so that: $P_1$ and $P_2$ are on ...
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3answers
47 views

Orthogonal circles

What is the equation of the circle that is orthogonal to the circles $x^2 + y^2 - 8x +5 =0$ and $x^2 + y^2 +6x +5 = 0$ and passes through the point $(3,4)$? I've spent hours trying to figure this out ...
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1answer
35 views

Proof there exists a point $O$ such $|AO|\le p,|BO|\le q,|CO|\le r$

Give the postive numbers $p=\dfrac{y+z-x}{\sqrt{3}},q=\dfrac{z+x-y}{\sqrt{3}},r=\dfrac{x+y-z}{\sqrt{3}}$,in $\Delta ABC$, where $|AB|=z,|BC|=x,|AC|=y$,and $G$ is centroids, I have prove following ...
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19 views

The largest sphere trapped into a tetrahedral void

A tetrahedral void is formed by four identical spheres each having a radius $R$ (as shown in the diagram above). A largest sphere, centered at the point 'O' having a radius $r_{max}$, is completely ...
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1answer
20 views

Lines cutting regions

15 lines are drawn in a plane such that 4 of them are parallel. a. What is the maximum number of regions into which the plane is divided? b. How many of the regions are finite(bounded)? a) The ...
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8 views

Show that the preimage of any circle or line in $\mathbb{C}$ under the stereographic projection is a circle on $\mathbb{S}^2$.

So this is an exercise in my complex analysis notes, which has been given the solution as follows (where I write $z = π(x_1,x_2,x_3) = \displaystyle\frac{x_1}{(1-x_3)} + \frac{ix_2}{(1-x_3)}$ for the ...
2
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2answers
51 views

Find the area of the circle with extern point

Points $A, B, C, D$ are on a circle such that $AB = 10$ and $CD = 7$. If $AB$ and $CD$ are extended past $B$ and $C$, respectively, they meet at $P$ outside the circle. Given that $BP = 8$ and $\angle ...
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2answers
42 views

Parallelogram Area

Given a parallelogram with two diagonals 6 cm and 8 cm. And the angle between them 120 degrees. Find the bases length and the parallelogram area. BD = 6 cm AC = 8 cm (AEB) = 120 degrees AB = ? ...
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3answers
53 views

What is the ratio of $\frac{XY}{SQ}$

In the picture,$PQRS$ is a prallelogram. $PS$ is parallel to $ZX$ and $\frac{PZ}{ZQ}=\frac{2}{3}$. Then$\frac{XY}{SQ}$ equals: The answer is $\frac{9}{40}$ Help me with the idea to solve this ...
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26 views

Find the intersection point of two tangent circles with given information

I am trying to find the tangent point between the two radii (R1 and R2) if only R1 is given and the points connected to the radii are given. I am trying to figure out an equation that is not an ...
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1answer
22 views

What can be said about quadrilateral BCED, and why ? (no other data than the drawing) [on hold]

What can be said about quadrilateral BCED, and why ? (no other data than the drawing)
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62 views

If the Greeks had been four dimensional, would they have been able to derive the pi squared coefficient for the hypersphere volume without calculus?

I was reading about Archimedes' pre-calculus proof of the volume of the sphere and I realized that the trick he uses (volume of hemisphere + volume of cone = volume of cylinder) doesn't generalize to ...
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22 views

Constructing Pythagorean Polygons

I ran into this idea of "Pythagorean Polygons" on a problem from Project Euler, and I thought of an interesting question. A "Pythagorean Polygon" is defined as a polygon that is cyclic and has its ...
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1answer
16 views

Regular polygon Interior angles

I am to find if any given angle(say x)can be interior angle of regular polygon.In other words,is there a regular polygon which angles are equal to X. I know the formula for sum of interior angles of ...
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1answer
32 views

Calculating semi-minor axis of an ellipse

I'm coding a solar system animation and so far it's done, but the the orbits of the planets are circular. To make the simulation more realistic, I want to use elliptic orbits. So I visited Mercury ...
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1answer
20 views

Simple vector algebra question for perpendicular lines

I'm actually a little ashamed to ask this, Suppose A and B are not 0. Consider a line "l" whose cartesian equation is $Ax+ By + D = 0$. Suppose that $P_0 = (x_0,y_0)$ does not lie on "l". Show that ...
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56 views

What is a 3-cell? What is a 3-disk?

Checking the usual places on the Web doesn’t (right now) yield a short answer to this simple question. I’m worried that I will spend just as long trying not to confuse $n$- with $(n+1)$-cells as I did ...
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1answer
15 views

How to test for two segments intersecting, excluding their endpoints?

Given two segments, for example S1=(2,1)-(2,3) S2=(7,8)-(2,3) the intersection test would be false. However, if I use the cross-products test the result is true. I understand the two ...
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1answer
23 views

Angles between points in $3$D space where the Origin is not the vertex.

Given two points $P_1,P_2$ in $3$D space that are positioned around a third point $M$, how do you calculate the angle between $P_1,M,P_2$. I know there are a few questions on here discussing how ...
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2answers
66 views

Find an Angle of a Right Triangle Without Trigonometric Functions

I have a right triangle triangle. I know the length of the hypotenuse (H) and one adjacent side (A). I would like to find the angle between the A and the H without using $\arccos(A/H)$. I would like ...
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5 views

Is the mapping achieved by the discrete Frechet Distance the best achievable pair decomposition?

I have the following proble: There are 2 sequences $P$ and $Q$ composed by $M$ and $N$ points respectively. I want to calculate the discrete Frechet distance between them. This operation outputs also ...
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1answer
24 views

Prove the Apollonius' theorem.

Let in a $\Delta ABC$, D is the midpoint of $BC$.Prove that: $AB^2+AC^2=2CD^2+2AD^2$ MY ATTEMPT : Given that $BD=DC$ and we construct $E \ such \ that\ AE=EC\implies AC=\frac{EC}{2} \ and \ ...
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39 views

Can i connect these points in a way that satisfies these conditions?

I apologise in advance for the horrible phrasing. If you imagine 3 points like this: You would then draw a path between A and B and another between B and C, such that the length of the line AB is ...
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19 views

Is a convex salient cone necessarily contained in an open half-space?

A cone $C$ in $\Bbb R^n$ is said to be salient if it does not contain any pair of opposite nonzero vectors; that is, if and only if $C \cap (-C) \subset \{0\}$. Obviously, a cone $C$ such that that ...
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6 views

How to clean Self intersecting Polygon ( remove the intersecting points ) [Multipolygon to single polygon]

I have been googling it almost for a week, my problem is that, I have a polygon which is made up of Latitude and Longitude points, I have used Douglus Pecker Algorithm to Decimate the polylines to ...
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1answer
137 views

What is the maximum volume of $N$-D slice of an $M$-D hypercube?

Consider a unit hypercube of M dimensions. We wish to make a cut of dimension N through it. What is the largest N-D size (length, area, volume, ...) we can achieve, $S(M, N)$, and what cut gives it? ...
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1answer
18 views

Length of a projected line

If a line is of true length x and is inclined in angles a,b,c with respect to the xy,yz,zx planes respectively , then how can i find the length of the projected line in the xy , yz and zx planes ...
2
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1answer
78 views

snugly fitted spheres in a cube [on hold]

A larger sphere A, having a radius $R$ is snugly fitted in a cube (i.e. sphere A touches all six faces of the cube). Further, a small sphere B is snugly fitted in the corner of cube (i.e. sphere B ...
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1answer
10 views

Finding end point of a segment, given start point and inclination

Consider a line segment whose start coordinates $(x,y,z)$ are known, and whose inclination $(a_1,b_1,c_1)$ in all $3$ planes is also known. The length of the line segment is known too. How do we find ...