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.
6,858 questions
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Proving that the trajectory of point $P$ such that $\frac{PF_1}{PF_2} = a$ is a circle of Apollonius
Let $F_1:(-c,0),F_2:(c,0)$ where $c>0$ be two distinct points on $x-$axis, and let $P$ be an arbitrary point on the plane.
As we know, the trajectory of $P$ is
$$\begin{cases}
\text{ellipse}, ...
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Geometry Problem About Tangent [closed]
Let C1:(x-y)²-8(x+y-2)=0 and C2:4x²+9y²=36 be two curves on x-y plane. If A,B be two common points on C1 from where perpendicular tangents are drawn to C2.
Then, if coordinates of A(a1,b1) and B(a2,b2)...
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For which integer n, is there a circle inscribing n gridpoints [closed]
I was browsing the internet and I found a question in German. I was not really satisfied nor really understood the anwser proposed there, and would like to get some further insight in this problem. I ...
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Contest Problem - Stanford Math Tournament 2024
I was trying to solve the problem from Stanford's math tournament and wonder if you could opine what is wrong in my approach: Problem No 13
https://www.stanfordmathtournament.com/pdfs/smt2024/team-...
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Locus of point of intersection of $x+y=m^2+m+1$ and $y-mx=1$, $m$ is a real number
Locus of point of intersection of $x+y=m^2+m+1\cdots (1)$ and $y-mx=1\cdots (2)$
I take $m=(y-1)/x$ from (1) and put it in (2), I get the locus as
$x^3-x^2+x-y^2+2y+x^2y-xy-1=0\cdots (3)$
But the ...
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Unable to know why angles have different values?
When drawing some line segments from the family $y=kx$, where k is real integer, I noticed that the lines $xk_i$,$xk_{i+1}$ have larger angle compared to pairs of lines when k is larger. To explain ...
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A rectangular hyperbola passes through the points A(1, 1), B(1, 5) and C(3, 1). The equation of normal to the hyperbola at A(1, 1) is A:
The solution states that the normal at A is parallel to BC. Could someone please explain why that is true. (the answer is 2x+y=3)
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Pair of lines joining origin to points of intersection of $y=2-x^2$ and $y^2=9x$
The parabolas
$y=2-x^2$ and $y^2=9x\cdots (*)$
can be checked to intersect at $P[\frac{3}{2}(3-\sqrt{5}), -\frac{3}{2}(1-\sqrt{5}]$ and $Q[\frac{3}{2}(3+\sqrt{5}),-\frac{3}{2}(1+\sqrt{5}]$ the Eqs. of ...
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Interesting Olympiad Locus Problem with coordinate bashing
Given circles $\omega_1$ and $\omega_2$. Let $M$ be the midpoint of line-segment joining both circles centers. Point $X$ and $Y$ on $\omega_1$ and $\omega_2$ respectively such that $MX$=$MY$.
Find ...
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Can Simmons' Submarine-Destroyer Problem be solved using geometry without ODEs?
George Simmons, in his classic book on differential equations, poses this problem:
A destroyer is hunting a submarine in a dense fog. The fog lifts for a moment, discloses the submarine on the ...
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A Question about the Product of the Slopes of Two Line Segments within a Hyperbola
Previously I have done a problem about the product of the slopes of two line segments within an $\textbf{ellipse}$, and I have posted here: Previous Post.
Before you read this current post, you can ...
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A Question about the Possibility of Transforming a Hyperbola into A Circle through Complex Geometry
I have been doing some questions where I can transform an ellipse
\begin{equation} \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1 \tag{1} , a>b>0, a>c>0\end{equation}
into a circle
\begin{equation} x^...
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A Question about What's Wrong with My Computation of Polar Equation of a Hyperbola Centered at its Right Focus
I am trying to compute the polar equation of a hyperbola which uses the right focus $F_2$ as the center of the coordinates, but the result I got is wrong. Here is my solution:
Given the euqation of a ...
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The product of the alternating lengths of a hexagon whose vertices belong to an ellipse
About 6 years ago, I came up with a wonderful ellipse property, and I did not prove it and forgot about it, but now I remembered it
I put my question here in order to get proof, and also to know if ...
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What would be the function y(r) of the y-coordinate of the center of a circle with radius r dropped on a cosine function at x=π?
Imagine dropping a circle with radius $r$ on a cosine wave at $x=\pi$, so it is tangent (and perpendicular) at two points. What would would be the y-coordinate of the center of the circle? One way to ...
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Restriction of a voronoi diagram to a plane?
Say I have a set of points $P$ in $\mathbb{R}^3$. Let $V = \{\mathcal{V_i}\}$ be the Voronoi diagram of $P$, where $\mathcal{V_i}$ are the Voronoi cells.
Assume we have a point $p \in P$ and assign ...
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enclosed equilateral triangle
Let $\triangle XYZ$ be an equilateral triangle and $A$ and $B$ variable points on segments $YZ$ and $XA$ respectively, and let $C$ and $D$ be the circumcenters of $\triangle XAY$ and $\triangle BAY$ ...
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Prove or disprove that the envelope of some chords of a conic section is another conic section
The problem is as follows:
Let $\Gamma$ be an ellipse and $A$ is a point in the interior of $\Gamma$. Let $B,C$ be two moving points on $\Gamma$ such that $AB\perp AC$. Prove or disprove that the ...
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Prove that $BD{\parallel }MN$.
In the figure below, $BC=CD$ and points $A$ and $P$ are arbitrary points on the circle.
Prove that $BD{\parallel }MN$.
So far, I tried to compare all equal arcs and subtended angles and also tried and ...
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Properties of special curves: cardioid, astroid, tractix, and so on
The question originated while trying to find coordinates of key points in a cardioid.
It is easy to find references listing points of interest and properties for conic sections online. Below are some ...
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Proving classically that positive reals $a,b,c$ satisfying the three aspects of the Triangle Inequality are lengths of sides of some triangle
Standard high school geometry courses cover the Triangle Inequality, which states that if $a,b,c$ are three sides of a triangle, then
$$a + b > c \qquad a + c > b \qquad b + c > a$$
I'm ...
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A Question about the Product of the Slopes of Two Line Segments within an Ellipse
$\textbf{Update:}$ I have posted a new question for similar property in hyperbola, and I give two analytic solutions to it. More thoughts, answers, comments are welcomed. Here is the link to New Post.
...
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Given the height, width, and number of segments of an equally segmented arc, find the length of the segment and the angle of the first segment.
Problem
A shape has a base of width $w$. There is an arc whose ends are concentric with the ends of the base. The arc is segmented (inscribed) by $n$ chords of equal length. The height of the shape (...
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Question to find points of intersection and altitudes with circumcircle and then multiply the three complex numbers
Let A(3,4), B(-4,3) and C(4,3) be the vertices of ΔABC. ΔАВС is inscribed in a circle S = 0 Let AD, BE and CF be the altitudes from A to BC, B to AC and C to AB respectively. AD, BE and CF are ...
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Local coordinates for a quadratic form
Suppose we have a real quadratic form in $v = (x_1, \, x_2, \, \cdots, \, x_n)$, which we WLOG can assume after an orthogonal change of coordinates to be
$$q(v) = -\sum_{i=1}^k x_i^2 + \sum_{i =k+1}^n ...
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Calculations using parabola
I've had quite a few calculations that can be done with parabola, so I think it's a good idea to collect them in one place, I'm asking this question maybe my list will be expanded, maybe reference ...
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Geometric mean in parabola
I just discovered a new feature using Geogebra, I want to know if it's already known or not, and I'd be happy to get many proofs.
$A,B$ are the lengths of any two parallel chords in the parabola, and ...
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Finding the equation of a line, knowing a relation among the distances to it from three vertices of a given triangle
I came across this problem in a book. In this problem, we are given three points A, B and C which represent the vertices of a triangle.
My try:
Assuming the line to be $ax + by +c = 0$, and because ...
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Why the split mass geometry trick to solve for ratios works in these problems?
I have an idea about "why" this center of mass trick OR mass point trick to solve for ratios in problems like this works
here suppose we are given any two ratios let the given ratios be AF:...
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Struggling to find an angle.
As the figure below shows, two triangles $ABC$ and $ACD$ are overlapping.
The question is asking what the value of angle $B$ is.
I drew the figure in $GeoGebra$ and found out that $AB=AC$ which means ...
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hyperplane intersecting a polyhedral cone
$\newcommand{\R}{\mathbb R}$
Consider a family $V$ of non-null vectors in $\R^n$. Consider the (open) polyhedral cone
$$C(V) = \{x\in\R^n\;:\;v\cdot x> 0\,,\;\forall v\in V\}\,.$$
We will call $C(V)...
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Is a curve on a finite interval always bounded?
Is it true that the image of a parametric curve $\gamma: I \subset \mathbb{R} \to \mathbb{R}^n$, where $I$ is one of $[a;b]$, $(a;b)$, $[a;b)$ and $(b;a]$, is always contained in some open ball with a ...
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Calculating maximum area between 2 curves
Find all possible values of $k$, for which area of the region bounded by the parabolas $2y^2=kx$ and $ky^2=2(y−x)$ is maximum.
My approach was first finding the point at which they meet and I got it ...
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Show a cubic surface containing $2$ skew lines is rational.
I am reading Shavalievich's book Basic Algebraic Geometry. In section 3.3 of chapter $1$, he gives an example to show that a cubic surface containing $2$ skew lines is rational. Here are the details:
...
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Deducing $\frac{a^2}{c^2} + \frac{b^2}{d^2} = 1$ from $h^2 +k^2 = c^2 + d^2$ and $\frac{a^2h^2}{c^4} + \frac{b^2k^2}{d^4} = 1$
Is there an elegant way to deduce
$$\frac{a^2}{c^2} + \frac{b^2}{d^2} = 1$$
given these equations?
$$\begin{align}
h^2 +k^2 = c^2 + d^2 \tag1\\[5pt]
\frac{a^2h^2}{c^4} + \frac{b^2k^2}{d^4} = 1 \tag2
\...
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Given a focus and directrix how can a series of ellipses be described?
Since an ellipse is the locus of points that has a constant ratio of distance between a focus (point) and a directrix (line), where that constant ratio is between 0 and 1, my conjecture is that given ...
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Difficult 3D geometry question
This is a past problem that came up in a high school entrance examination. Question 2 asks for the area of the trapezoid PQMN. Even after reading the answer sheet, it still doesn't make sense (ex. why ...
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find the area of the quadrilateral between the two coordinate axes and two intersecting lines, [closed]
Today someone asked me to solve a question related to calculating the area between the two coordinate axes and between two lines that have secant by knowing the coordinates of the points of ...
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Planes attached point by point produce plane bundles
The issues I met came from quaternionic functions, but since the algebra is not involved I would rephrase it considering only the linear structure $\mathbb{R}^4$ or $\mathbb{R}^8$. I would like to ...
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how to measure angular accuracy in x/y dimensions?
Imagine a coordinate system where the x axis is left-right, y is up-down, and z is straight ahead in front of me (or in front of the camera, to be more precise). Suppose I want to characterize a unit ...
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finding vertices of a rhombus given the equations of a side passing through a point and a diagonal
considering a rhombus ABCD,
sides AB and AC are represented by the equations $x-y+1=0$ and $2x-y-1=0$ respectively, if the line BC passes through the point E=(5,-6). find the equation of the sides BC, ...
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How to obtain the orthogonal basis of an ellipsoid section
Given an ellipsoid equation of the form \begin{equation} \textbf{v}^T \textbf{A} \textbf{v} \; = \; 1 \tag{1} \end{equation} where $ \textbf{A} \in \mathbb{R}^{n \times n} $ is positive definite and ...
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Projection that varies point by point is an open map
Let us consider the vector space $\mathbb{R}^6=\mathbb{R}^3\times\mathbb{R}^3$ and let us represent any vector as $(x,y)$, with $x,y\in\mathbb{R}^3$. For fixed $x\in\mathbb{R}^3\setminus\{0\}$, let $y^...
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The radius of a circle is equal to the distance between the focal points of the parabolas only if the two directrices are perpendicular
Today I came up with many beautiful results in parabola, but I decided to present the most prominent result in the form of a question on this site, I think it is a really special feature
Let's have ...
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Confusion about logical jump in proof relating to points on conic and cross ratio
Let $p$ and $q$ be two distinct points in $\mathbb{RP}^2$. Let $ L_p $ and $ L_q $ be the sets of all lines that pass through $ p $ and $ q $, respectively. Let $ \tau : L_p \to L_q $ be a projective ...
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Given $A,B,C$ in $3D$ space, find point $P$ such that $\angle APB , \angle BPC, \angle CPA $ are equal to set values
I am given three points $A,B,C$ in $3D$ space. I want to find point(s) $P$ such that $\angle APB = \theta_1$, $\angle BPC = \theta_2$, $\angle CPA = \theta_3$, where $\theta_1, \theta_2, \theta_3$ ...
user1431065
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Finding the ratio at which diagonals divide each other, in quadrilateral $ABCD$ with vertices $A(1,5)$, $B(4,1)$, $C(7,5)$, and $D(4,9)$
Find the ratio in which the diagonals of the quadrilateral $ABCD$ divide each other if the vertices are $A(1,5)$, $B(4,1)$, $C(7,5)$, and $D(4,9)$.
I realize that if I find the equations for the ...
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What is the locus of the centers of all spheres that are tangent to three given spheres having different radii
Given three spheres of radii $r_1, r_2, r_3$, and centered at $A,B,C$ in $3D$, what is the locus of all spheres that are tangent to these three spheres?
My Attempt:
Let $A = (x_1, y_1, z_1), B = (x_2, ...
user1431065
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A geometric interpretation of an algebraic equality
There is an peculiar algebraic equality:
$$\int_0^1\frac 1{x^x}\mathrm dx=\sum_{n=1}^{+\infty}\frac 1{n^n}.$$
Some context.
The proof is relatively straightforward, one can write $\frac 1{x^x}=e^{-x\...
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Are the usual spherical coordinates $(\theta,\phi)$ not a "true" local coordinate chart on the 2-sphere?
I've been self-teaching myself differential geometry and still somewhat new. As I understand it, a local coordinate chart on an $n$-dimensional, real, smooth manifold, $M$, is a pair $(M_{(q)},q)$ ...
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