Questions on the use of algebraic techniques to prove geometric theorems.

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Surface area of a section of the unit sphere

Let $v$ be a vector on the unit sphere in $\mathbb{R}^n$ and let $S(\epsilon)$ be the set of vectors $s$ on the same sphere such that $$ |s \cdot v| \leq \epsilon.$$ What is the surface area of ...
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High School Geometry problem with a triangle and trapezoid in a larger triangle.

I school, I have an assignment to write a problem for geometry students. I have written the following problem. Draw triangle ABC. Let the height have magnitude h. Draw a line segment, DE, which is ...
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20 views

Determine the value of y so that two line segments are parallel

Determine the value of $y$ so that the line segment with endpoints $P(3, y)$ and $Q(-3, -1)$ is parallel to the line segment with endpoints $R(-4, 9)$ and $S(5,6)$. I began by finding the slope ...
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Trigonometrical functions and complex numbers

(This question will at first appear too broad. However, the overall philosophy will be explained below in a way that asks specific questions, which I hope will be conducive to this being a reasonable ...
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2answers
56 views

Rational parametrization of circle in Wikipedia

In http://en.wikipedia.org/wiki/Circle but also in the corresponding article in the German Wikipedia I find this formulation ( sorry, I exchange x and y as I am accustomed to it in this way ) : "An ...
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doubt in proving Ratio.

Find the equation of the tangent of the curve $x^4+y^4=a^4$ at $(h,k)$. And prove that the point of contact of divides the line segment joint the intercepts of the tangent in the ratio $h^3:k^3$. ...
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How to calculate the solid angle of a spherical rectangle from astronomical angles

Say I have 2 astronomical angle pairs defining a confined region on the visible hemisphere: (minAzimuth, minElevation) & (maxAzimuth, maxElevation) How can we calculate the solid angle of the ...
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The sum of the abscissae of the intersections of a cubic and a line

I remember being told in passing in a talk once the following theorem: Let $y=x^3$, and let $x_1,x_2,x_3$ be the abscissae ($x$ co-ordinates) of three distinct points on this cubic. Then ...
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Intersection of 3 positively sloped planes

Suppose I have three planes, each of which is 'positively sloped' in the sense that the first plane intersects the x-axis at a positive value, and the y and z-axes at a negative value. Similarly, the ...
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Barycentric Coordinates of Orthocenter question

this page describes the barycentric coordinates of the orthocenter as $(\tan A : \tan B : \tan C)$. How would you prove this using the areal definition of barycentric coordinates? Thank you. EDIT: ...
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Affine transformation that sends a conic to itself but does not preserve the focci or the axes [closed]

So I'm trying to find an affine transformation that sends a conic to itself but does not preserve the foci or the axes. I don't know if this can be done. I'm pretty sure that if it is possible then I ...
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Vector Calculus Question- Planes and Curves

Will you please help me in the following? Let $\pi$ be a plane perpendicular to the curve: $$ \gamma(t) = (5\cos t, 5\sin t,-2t) $$ at the point $(x(t_0), y(t_0 ) ,z(t_0)) $ . We also know the ...
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Locus of complex numbers.

Question Let $P(x,y)$ be the point on an Argand diagram representing the complex number $u=x+iy$ and satisfying the equation \begin{align*} \vert u \vert=k\vert u+a\vert, \end{align*} where $k$ is ...
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6 views

Books on vector analytic geometry

I'm looking for books about analytic geometry which covers affine change of coordinates,equivalence of conics by affine and projective change of coordinates, etc. using vectors. I would like a book in ...
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1answer
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On the associative property of a binary operation of the fundamental group.

I was reading about the proof of associativity property of the operation on the fundamental group here. The book gives the following diagram then it says the reader should supply the elementary ...
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1answer
18 views

Question regarding condition of perpendicularity

Let $ax^2+2hxy+by^2=0$ be the equation of two straight lines passing through the origin. We know that the angle between these two straight lines is given by, $$\arctan \dfrac{2\sqrt{h^2-ab}}{a+b}$$ ...
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1answer
76 views

Complex hypersurface globally defined

Let $A$ be a pure one-codimensional analytic subset of a domain $D \subset \mathbb{C}^n$. Is it true that $A$ is defined by one single holomorphic equation $f(z)=0$ if $D$ is bounded and pseudo-convex ...
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I'm having troubles to find this parametrization.

I'm reading the Reid's Undergraduate Algebraic Geometry book of algebraic geometry for undergraduates and I have two questions about a proof of an example on the page 19: Red question: Reid said ...
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Prove that the intersections of the ray $f(x)\rightarrow x$ with the $n$-disk form a continuous function, with $f$ continuous

This appeared in a proof of the Brouwer fixed point theorem, in Introduction to Algebraic Topology, by Rotman, but it was left as an exercise. I could only prove this in 2 and 3 dimensions, not in ...
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Get angle in degrees of coordinate on circle.

So assume I have coordinates of two points on a circle, and the coordinate of the center of the circle. How would I go about finding the angle of the points in degrees?
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Bound for the distance of projections onto the unit sphere

Given $x \in \mathbb{R}^n$, $x \neq 0$, let $x' = x/|x|$ (where $|\cdot|$ is the euclidean length) be its projection onto the unit sphere. I would like to prove that $$ |x' - y'| \leq 2 ...
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How do I prove that the set of all possible points of R lie on a circle? [on hold]

$P(-1,11)$, $Q(2,2)$ and $R(x,y)$ are points that $|PR|=2|QR|$. Find the set of all possible points of $R$ lie on the circle. (Find locus)
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finding a point on a surface? the surface is an ellipsoid

I have drawn the cross-sections of the surface $2(x-1)^2 + (y+2)^2 +z^2 = 2$ for the given planes, but am now asked to write down a point which is on the surface. I have no idea how to go about this, ...
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1answer
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Reference to line parametrization

Defining two lines in space, $\mathbb{R}^3$, as: $l_1: \textbf{a}_1+\lambda_1\textbf{b}_1$ $l_2: \textbf{a}_2+\lambda_2\textbf{b}_2$ The line to line intersection condition is: $\textbf{b}_1\cdot ...
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Geometry involving area of rhombus and interior isosceles triangles

Points E, F, G, and H lie inside a rhombus ABCD, such that the triangles AEB, BHC, CGD, and DFA are isosceles right triangles with hypotenuses AB, BC, CD, and DA.The sum of areas of ABCD and EFGH is ...
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Prove that $(A,B)\sim(P,Q)$ and $(C,D)\sim (P,Q)\implies (A,B)\sim (C,D)$?

I have the following laws: And I did the following: $(A,B)\sim(P,Q)\wedge (C,D)\sim (P,Q) \stackrel{?}{\implies} (A,B)\sim (C,D)$ $(A,B)\sim(P,Q)\wedge \stackrel{symmetry}{(P,Q)\sim ...
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How to calculate a reduced volume?

Let's say we have an irregular 3D shape with volume=V ( we know V but we don't know its equation= F). Now I want to calculate another 3D shape which is exactly the same shape but one size smaller, ...
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R. Blum equation of tangents clarification.

In Coxeter's Intro to Geometry, exercise 4 pg 114 restates a finding in Richard Blum's paper. On page 2, where he introduces the equation of the tangent lines: T(xi,eta)*T(xi0,eta0) - T^2(xi,eta | ...
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3D vector perpendicular calculation

Three points $A(6,7,-6)$,$ B(0,0,0)$ and $C(2,6,9)$ are given which are the vertices of a cubes. Find the coordinates of another vertex not on the $ABCD$ plane. I found the answer by finding the ...
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Vector on bisectrix between other two

Supose $\overrightarrow a=(2,-3,6)$ and $\overrightarrow b=(-1,2,-2)$ are represented in the same origin. Calculate the coordinates of the vector $\overrightarrow c$ that is on the bisectrix of the ...
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Find a point on the same alignment of normal vector of a plane

I need to find a point(x,y,z) that is - distance 2 from a known point P (x1,y1,z1) - on the same alignment of normal vector for plane A - P is on the plane A the same question as: Find a point ...
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Fitting a circle

Given a figure like , how can I determine the radius of the circle with middlepoint H analytically? CDFE is a square with sides 6/5, with E and F being points on the circles with radii 2.
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Prove that $|AM|=|BM|$, if …

Let $M$ be the point of intersection of the diagonal sides of a trapezoid. Let $l$ be the line through $M$ that is parallel to the bases of the trapezoid. Let $A$ and $B$ be the points in which the ...
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$\partial (Q A) = Q (\partial A$) for an orthogonal matrix $Q$?

Let $A \subset \mathbb{R}^d$ and let $Q \in \mathbb{R}^{d \times d}$ be an orthogonal matrix. For a set $B \subset \mathbb{R}^d$, denote $Q B:= \{ Qx : x \in B\}$. Does it hold for the boundary of the ...
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Help with this coordinate geometry question involving cirlces and parabolas.

Question: A point $P$ in a plane moves such that it remains at a fixed distance $r$ from a fixed point $A\equiv(r,r)$. (i) Find the equation of the locus of point $P$ (in terms of $r$). Another ...
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Algebraic calculation steps.

Can somebody explain how the coefficients $a_{11}, a_{12}, a_{22}$ are derived after rotating the ellipse below ?? $\widetilde{s_{11}} = \frac{\sum_{j=1}^n(x_{jk} - \bar{x_k})}{n}$ Thank you in ...
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Is rotation in $\mathbb{R}^d$ unique?

Let $\boldsymbol{u} \in \mathbb{R}^d$ such that $||\boldsymbol{u}||_2 = 1$ be a directional vector. Let $Q_{\boldsymbol{u}} \in \mathbb{R}^{d \times d}$ be an orthogonal matrix such that ...
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Lattice points-Triangle

We have a triangle $ T $ with vertices at the $ \mathbb{Z} \times \mathbb{Z} $ grid . Now, consider the surface $ 2T= \{x \in \mathbb{R}^2 : \frac{x}{2} \in T \} $ ( so, double $ T $ ). Is it possible ...
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What is the reason behind the Pythagorean relation in a hyperbola?

I am currently (in my Pre-Calculus course) deriving the equations of the conic sections. I very much understand how the relationship, in an ellipse, between $a, b$, and $c$ is established. Knowing ...
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Aspect Ratio of Cylinder, Pyramid and Dome

The aspect ratio can easily be defined for rectangular geometries ($AR = height/width$). Is there a definition for aspect ratio of a dome, cylinder, and pyramid (Here standard pyramid and dome were ...
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2answers
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Locus of image of point in a line.

I am given the following question: Find the locus of the image of the point $(2,3)$ in the line $$\text{L}:(2x-3y+4)+k(x-2y+3)=0$$ where $k$ is any real number. Attempt at solution. I ...
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1answer
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Coordinate-geometry curiosity question

How can we draw a triangle give one of its vertex and the orthocentre and circumcentre? I tried to invoke the concept of 9 point circle and tried using the centroid but could not succeed in making ...
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1answer
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Find the intersection of two lines passing through given points

Line A goes through the points (4,5) and (-2,-1) and line B goes through the points (3,3) and (6,1). At what point do they intersect? I found the equations of the 2 lines, for A I got: $y = 9-x$, ...
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How can one solve this equality geometrically?

Given that $x,y$ are real such that: $$x+2y=\dfrac{1}{2},$$ how can one show, geometrically that $$x^2+y^2\geq \dfrac{1}{20}?$$ I see that $x^2+y^2-\dfrac{1}{20}=5\left(y-\dfrac{1}{5}\right)^2$ ...
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Analytic-geometry rotation concept

I am confused how my book comes up with the following formula- Lets consider a Right angled Isoceles triangle with $2$ vertices on hypotenuse given as $(x_1,y_1)$ and $(x_2,y_2)$ Now the 3rd ...
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1answer
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What are the distances from a line to the tangents of a circle?

I have a line given by two points, and a circle given by its origin and radius. I need to find the perpendicular distance between the line and the two tangents of the circle that are parallel to the ...
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1answer
51 views

Find the focus, vertex, latus rectum of the parabola

The problem is, Find the focus, equation of directrix, vertex, length of latus rectum of the parabola given by, $$\left(\alpha x+\beta y+\gamma\right)^2=Ax+By+C$$ I am stuck with the problem for ...
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Tangents are drawn from any point on a hyperbola, to a circle. Find the locus of mid points of the chord of contact.

Tangents are drawn from any point on the hyperbola $\dfrac{x^2}{9}-\dfrac{y^2}{4}=1$, to the circle $x^2+y^2=9$. Find the locus of mid points of the chord of contact. Attempt:- Taking the point to be ...
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3answers
449 views

Why this polynomial represents this figure?

I'm trying to understand why this figure is represented by this polynomial expression: My goal is to prove directly why cartesian product of natural numbers is equinumerous to the natural numbers. ...
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Normal Vector of plane for Rotation

I reading a code where Normal vector to a plane is given. then a,b,c are taken (what I guess is direction ratio values). ...