Questions tagged [analytic-geometry]
Questions on the use of algebraic techniques for proving geometric facts. Analytic Geometry is a branch of algebra that is used to model geometric objects - points, (straight) lines, and circles being the most basic of these. It is concerned with defining and representing geometrical shapes in a numerical way.
7,008 questions
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Is it possible to have a regular pentagon with all integer cordinates? [closed]
Intuitively, I think yes, but how?
If the answer is really yes, which is the smallest one?
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Why does the leading coefficient of a quadratic trinomial resemble some sort of a slope?
It's known that a unique parabola of the form $y=ax^{2}+bx+c$ exists for any three distinct points, provided that the points are non-collinear and their $x$ coordinates are distinct.
Consider the ...
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Generic and Special Fibres for Rigid Analytic Model $\Bbb G_{m}^{\text{an}}$ of Multiplicative Group
An absolute beginner level question on rigid analytic spaces & their interplay with formal schemes (after Raynaud). In this minicourse by Bosch on this topic he introduced as motivation following ...
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Can it be proven that CVT(Continuously Variable Transmissions) are impossible using only smooth rigid bodies?
In mechanical engineering, there are many designs of a CVT(Continuously Variable Transmissions) that is able to change gear ratio continuously, but all of them are unsatisfactory for some reason.
...
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Inscribing a rhombus inside a triangle (Pythagorea 22.17)
Recently I stumbled on a game called Pythagorea. The idea is that you have a geometric challenge, in this case "Inscribe a rhombus inside the given triangle, such that they share the common angle ...
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Circle on the Argand Plane
So the question looks simple enough but I am having troubles to come up with a solution. Let $\theta$ be the argument of $w$, then we have $w=|w|e^{i\theta}, z=|z|e^{2i\theta}$. So it suffices to ...
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Reflection of a Curve about a Line
so I was going through the following proof on how do we derive the new coorinates of the point on the reflected curve about a Line, and I couldn't understand one thing,How can we write $ x_f = x_0 + ...
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Locus of the foci of all the conics of given eccentricity
In W. H. Besant's classic book Conic Sections: Treated Geometrically there is a question at the end of the first chapter:
Find the locus of the foci of all the conics of given eccentricity which pass
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What makes one p-adic isometry be rational preserving, and another not?
What makes one p-adic isometry rational-preserving, and another not?
Consider the function $f(x)=\dfrac{ax+b}{cT(x)+d}$ where $a,b,c,d$ are 2-adic units.
Definition: A rational-preserving 2-adic ...
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Existence of four concyclic points on the graph of a real function under scaling of abscissa
Suppose $f:\mathbb R \to \mathbb R$ is a smooth function.
Fix four distinct reals $u_1<u_2<u_3<u_4$. For each $x_0\in\mathbb R,\lambda\in\mathbb R-\{0\}$, define four points
$$
\bigl(x_0+\...
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The set of centers from which some circle has $4$ intersections with $y=x^3-ax$ expands to cover the whole plane as $a → ∞$
For all $a\in\Bbb R$ let $S_a$ be the set of centers from which some circle has $4$ intersections with the graph of $y=x^3-ax$.
For example, in the image, $(1,0)$ is the center of a circle which has $...
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The parabola passing through intersection of lines $2x+3y+1=0, 3x+2y-1=0$ with the circle $x^2+y^2=4$
$S'=S+t~ L_1 L_2=0, t\in \mathbb R $ represents a family of conic sections passing through intersection points of the conic $S(x,y)=0$ with the lines $L_1(x,y)=0$ and $L_2(x,y)=0$.
Here, in this ...
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Equation of the quadratic curve if two tangents and corresponding chord of contact are given
Given a quadratic curve/conic $S(x,y)=0$ and an outside point $(x',y')$ we can get equation of chord of contact $T(x,y)=0$ and the combined equaion of the corresponding tangents as $$T^2(x,y)=S(x,y) S(...
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A property of the rectangular (isosceles) hyperbola
A few days ago, while experimenting with GeoGebra, I observed the following property:
If we have a rectangular hyperbola (i.e. an isosceles hyperbola with perpendicular asymptotes) with vertices $A,B$,...
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Find the intersection of the lines depending on a parameter k [closed]
My teacher has asked us to solve these exercises and I don't know what to do. I know what to do when the equations of the lines do not have a parameter.
Discuss depending on $k ∈ ℝ$ the intersection ...
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Semicircle, inscribed circle and parabola all tangent to each other
I have the following set-up:
A semicircle of radius $R = 1$, centered in $(1,0)$ described by the equation $$i)\qquad (x-1)^2 + y^2 = 1, \quad y\ge0$$
A parabola with equation $$ii)\qquad y=-ax^2+bx$$ ...
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Terminal side in polar coordinates
In "Calculus With Analytic Geometry" , the author (G.F. Simmons) says the following, while explaining Polar Coordinates:
Distance is given by the directed distance $r$, measured out from ...
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Difference in parabola and upper part of hyperbola
In a book for beginner undergraduates, conic sections are introduced as sections of a cone by a plane. Then their examples are shown in terms of picture.
In the real-life examples, the teacher ...
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Locus of the incenter under perimeter and side constraints — three conic-related theorems and a puzzling exception
Yesterday, while working with GeoGebra, I stumbled upon three beautiful and seemingly related geometric theorems involving the incenter of a triangle under different constraints. I would like to share ...
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Estimating the number of lines with slope of rational number
This is Exercise 7.2 of Polynomial Methods in Combinatorics:
Let $S_r$ be the first $r$ rational numbers in the sequence $0, 1, 1/2, 1/3, 2/3, 1/4, 3/4,...$ Let $G_N$ be the $N \times N$ integer grid....
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Can Playfair's Axiom be generalized to an arbitrarily large dimension?
Playfair's Axiom reads: "There is at most one line that can be drawn parallel to another given one through an external point." It's equivalent to the parallel postulate in 2d, but I'm not ...
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Why is the rationality of nondegenerate conic bundles a corollary to Tsen's theorem?
Let $X \subset \Bbb{P}^2\times \Bbb{A}$ be the conic bundle defined by $q(x_0:x_1:x_2;t) = \sum_{i,j = 0}^2 a_{ij}(t)x_ix_j$ ($k$ is algebraically closed). If $\det | a_{i,j}(t)\mid$ is not ...
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Constructing the function of a "cross section" produced by a rotated plane cutting the graph of any given 3d function
Apologies for the function redundancy. I am having trouble finding if the general construction of the function of the "cross section" of any 3d function's graph cut at arbitrary angles has ...
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Show that two of these points are separated by a distance of at least $ r^{1/3} $?
Three distinct points with integer coordinates lie in the plane on a
circle of radius $ r > 0 $.
Show that two of these points are
separated by a distance of at least $ r^{1/3} $?
I found this ...
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How to add equal offset to an ellipse at all directions?
The basic formula of an ellipse is: $\frac{(x - cx)^2}{rx^2} + \frac{(y - cy)^2} {ry^2} = 1$
In my case, the ellipse is centered at $(0, 0)$, so the formula becomes
$$
\frac{x^2}{rx^2} + \frac{y^2}{ry^...
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