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

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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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1answer
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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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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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66 views

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
19 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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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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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
25 views

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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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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1answer
24 views

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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71 views

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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2answers
37 views

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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133 views

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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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
60 views

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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1answer
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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
72 views

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
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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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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). ...
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1answer
42 views

Orthocentre of a triangle

How do we determine the orthocentre of a triangle when the vertices are given as $(0,0),(x_1,y_1),(x_2,y_2)$? In a normal case i would take out the equation of any two perpendicular bisectors, get ...
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2answers
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Determining the 3rd vertex of an Equilateral and Right angled isoceles triangle.

I am really having problems solving the following problems: If $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of the two vertices on the hypotenuse of a right angled isosceles triangle then the ...
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3answers
40 views

What will be the other vertex of the triangle?

Two vertices of a triangles are $(5,-1)$ and $(-2,3)$. If the orthocenter of the triangle is the origin, what is the other vertex ? My approach was that since the three vertices and the orthocenter ...
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Can area be irrational?

I'm stuck in a question of my book which says: If in an equilateral triangle the coordinates of two vertices are integral then what can we say about the coordinates of the third? The answer is that ...
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Graphical transformation

I have a burning question to ask regarding graphical transformation: Suppose I have a function $f(x)$ I want to find $f(ax+b)$ for non zero $a,b$. There are two approaches that I can go: First: ...
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Can a line in 3-space have all direction cosines $=\frac{1}{2}$

I immediately found that it is impossible since the squares of the direction cosines have to add to 1 and $3 \times (\frac{1}{2})^2 \neq 1$. However, the textbook asks to "interpret geometrically", ...
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Geometrical properties of tetrahedra under rotation

Consider two tetrahedra which share the same point of origin but differ in both scale and rotation over the X-axis. Can someone explain why the following points meet with these parameters? Both have ...
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How to plot quadratic forms

I'm studying quadratic forms in algebra at the moment and I've been asked to plot the following curve: $$3x^2+4xy+3y^2-\sqrt{2}x+\sqrt{2}y=1$$ I have used the following transformations: ...
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Conics - required to show $SR \times S'R' = b^2$

Consider the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b > 0$. $R$ and $R'$ are the feet of the perpendiculars from the foci $S$ and $S'$ on to the tangent at ...
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Pascal and Brianchon's theorems for hyperbolic paraboloid

How would one formulate a version of these two theorems for the hyperbolic paraboloid, and what would be a simple proof? How are the classical formulations of these theorems related to this quadric?
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Prove quadrics are rational algebraic surfaces.

I have to prove that an irreducible quadric in RP^3 is a rational algebraic surface, ie, the homogeneous coordinates of any point can be expressed as polynomials in two variables. My idea was to do ...
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Quadratic singularities and local curves.

Everything is to be understood over the complex field. Assume you have two finite dimensional $\mathbb{C}$-vectorial space $V$ and $W$. You are given a bilinear form : $$\phi:V\times V\rightarrow W ...