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.
7,013 questions
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Determine the reflection line about which two points reflected will lie on two given lines respectively
You're given two lines in the $xy$ plane, let's say
$ Line 1: a_1 x + b_1 y + c_1 = 0 $
and
$ Line 2: a_2 x + b_2 y + c_2 = 0 $
In addition you're given two points $P = (p_1, p_2) $ and $Q = (q_1, q_2)...
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Congruent triangle to a given one, with its vertices at specified distances from the original triangle vertices
You're given $\triangle ABC$ with known vertices in the $xy$ plane. The coordinates of $A,B,C$ are known. Now, given three distances $d_1, d_2, d_3$. You want to determine all congruent triangles $\...
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Loci of sliding ellipse foci [duplicate]
An ellipse of major axis and eccentricity $(2a,e) $ slides up and down contacting the coordinate axes $ (x,y)$ always.
What are the loci of individual foci?
At any instant the variable pentagon has ...
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Find the ratio of side lengths of two equilateral triangles given a midpoint condition
Problem Statement:
As shown in the diagram below, we have two equilateral triangles, $\triangle ABC$ and $\triangle ADE$, sharing a common vertex $A$.
We construct a line connecting vertices $B$ and $...
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Similarity argument in right triangle with perpendiculars to sides
I have a right triangle $OAB$ with right angle at $O$, and let
\begin{equation}
OA = L, \quad OB = 1.
\end{equation}
Let $a$ be a point on $OA$ and $b$ a point on $OB$. From these points, ...
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Perpendiculars passing through diagonal intersection in a quadrilateral formed within a square
Let $ABCD$ be a square with points $F \in BC$ and $H \in CD$ such that $BF = 2FC$ and $DH = 2HC$.
Construct:
Line through $F$ parallel to $AB$, meeting $AD$ at $E$
Line through $H$ parallel to $BC$, ...
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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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Proof of the wheel stud problem
Consider a regular $n$-sided polygon in two-dimensional space (metaphorically: the stud pattern on a typical car wheel), and a sequence to choose each of its sides exactly once. If the choice sequence ...
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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 ...
- 183
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Are analytifications in Berkovich geometry strict k-analytic spaces?
Let $k$ be a non-archimedean complete field. Berkovich defines the analytification of a $k$-schemes $X$ which are locally of finite type via the assignment
$\mathrm{Spec}(A) \mapsto \mathcal{M}\mathrm{...
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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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Find the ratio of $BA:AC:CB$ if a certain point $D$ on triangle $\triangle ABC$ satisfies a ratio
Let $D$ lie on side $AB$ of right-triangle $\triangle ABC$ such that $AD:CD:DB = 3:2:1$
Then, find the value of $BA:AC:CB$
The only method I could think of would be to use coordinates where $C=(0,0), ...
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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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Finding the points on a cone where the tangent plane contains the line formed by intersecting two given planes
There are several resources (especially Calculus books) that talk about finding tangent planes to a given (level surface) with certain prescribed conditions. I was trying to make a general question ...
- 4,187
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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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what does analytically isomorphic mean for complex spaces？
Here"analytically isomorphic"means that the completion of the local rings of two points in some complex spaces are isomorphic. The smooth case is trivial so the only interesting case is that ...
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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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Maximum area shape that can pass through concatenation of constant curvature channels
Which shape fits inside and can completely pass through a channel defined by a two dimensional curve with thickness $t$ (its walls are at distance $±t/2$ from the centerline) and consisting of a ...
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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 + ...
- 3,478
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What is the correct definition of "double tangent"?
In the literature , there are two definitions of "double tangent".
One of these is a single straight line which is tangent to a curve at two different points.
I am not worried about this -- ...
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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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Confusion about the "dual" map between Archimedean and Catalan solids
From what I've read, the Archimedean and Catalan solids are "dual" shapes. From an algebraic/graph-theoretic standpoint this is easy to understand:
The number of faces in one Archimedean ...
- 4,667
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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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Let $A\equiv (3,5,4)$, $B\equiv (4,3,5)$ and $P\equiv (a,b,0)$. If P be such that $\angle APB\in[0^{\circ},180^{\circ}]$ is maximum, find $a$ and $b$
Let $A\equiv (3,5,4)$, $B\equiv (4,3,5)$ and $P\equiv (a,b,0)$.
If point P be such that $\angle APB\in[0^{\circ},180^{\circ}]$ is maximum,
then find the value of $a$ and $b$.
My Attempt:
If $P$ lies ...
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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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Existence of circle passing through four distinct points on $y=x^a,x>0$
I’m wondering if four distinct points can lie concyclic on the graph of $y=x^a,x>0$.
For $1/2\le a\le 2$ or $a<0$, no such circle exists, as the determinant’s sign is stable.
For $0<a<1/2$...
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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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Parabola touching the circle $x^2+y^2=4$ and the line $3x+4y=10$ at the same point
$S'(x,y)=S(x,y)+t L^2(x,y)=0, t \in \Re$ represents a family of conic sections which touch the conic $S(x,y)=0$ and the line $L(x,y)=0$ at the same point.
For example, the parabola touching a circle $...
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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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Circle passing through $(3,4)$ and touching $x+y=3$ at $(1,2)$
See
Family of Circles Touching a Circle and a Line
$S'(x,y)=S(x,y)+t L(x,y)=0, t\in \mathbb{R}$ is a family of conics passes through the conic $S=0$ and cutting /touching the line $L=0$.
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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When is most of an inner ball outside the hypercube which bounds the outer balls which bound the inner ball?
There is a simple question where a unit square contains four circles of radius $\frac14$ with a fifth inner circle, and the aim is to find that the radius of the inner circle is $r_2=\frac{\sqrt{2}-1}{...
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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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Arithmetic and geometric mean of numbers which can be written as the sum of two squares
Let $A_n$ and $G_n$ be the arithmetic mean and the geometric mean respectively of the first $n$ numbers
$$
1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32
$$
which can be written as the sum ...
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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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What is the general form for solving for the count of possible connections between a given number of points?
I can do this by hand, but it will take forever. I'm not 100% sure how to phrase the question, and I'm not a mathematician, but I'll do my best.
Any two points can be connected by exactly one line, ...
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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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All values of $t$ so that origin lies in the obtuse sector of the lines: $4tx-2ty+3-t=0, 3x+ty+t-1=0$ [closed]
What can be a good approach to find
all values of $t$ so that origin lies in the obtuse sector of the lines: $$4tx-2ty+3-t=0 ~\&~ 3x+ty+t-1=0.$$
Please help.
Edit: Following @Z.Ahmed, I get:
$$\...
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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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Minimum Perimeter of triangle with co-ordinate axes as two sides and the third side passing through the point $(8,1)$
A straight line with a negative slope passes through the point $P(8,1)$ and meets x-axis at $A$ and the y-axis at $B$.
Find the minimum possible value of the perimeter of $\triangle AOB$.
My Attempt
...
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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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How the angle between the projection of final y-axis and the initial z-axis is $\gamma$ in XZX (intrinsic) convention of Euler Angles?
I was working with rotating a frame to another frame recently, and got to know about Euler Angles. Since I had to find the specific rotation angles for a given final set of $X$-$Y$-$Z$ axes, I got ...
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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 ...
- 4,461
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$|F(x)|=|F(x)-x|=2|x|$ for $x$ in $\mathbb{R}^n$. Prove that $x, F(x), F(F(x))$ are linearly dependent vectors.
Let $x$ be a vector in $\mathbb{R}^n$, $F$ is a linear transform. It is given $|F(x)|=|F(x)-x|=2|x|$ for this vector. $|x|$ is the length of the vector. Prove that $x, F(x), F(F(x))$ are linearly ...
- 301
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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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Is it legitimate to alter the spherical coordinate system?
Is the only reason the polar angle is measured from the north pole to the south pole (0$^\circ \leq \phi \leq 180^\circ$) by convention so that it has no negative measurement values ?$\;$ I find ...
- 263
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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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Where is the inconsistentency hiding in Langley's adventitious angle?
This question was in my maths book and there were 4 options given 30°, 40°, 20° and 25°. At first I tried to eliminate the options because I thought as there could be only 1 answer so all other angles ...
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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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Equation of a line of reflection of two points in another point
Three points with known coordinates are given - $P_1 = (x_1,y_1)$, $P_2 = (x_2,y_2)$ and $P_3=(x_3,y_3)$. The line $y_l(x)$ goes through a point $P_2$. A point $P_a$ is an intersection point of lines $...
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