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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.

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100 views

Convergence of the sequence of the orthocenters, incenters and centroids of a triangle.

Now asked on MO here. Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, $B_{n+1}$ is ...
2 votes
0 answers
23 views

Determining the Plane for a Given Point and Line Using the Equation of a Plane

I was working on an exercise following an example that required exactly what I had to do as well, but I encountered some difficulties. Below, I have indicated 2 titles, the first one is the example I ...
3 votes
2 answers
159 views

Find the radius of the red circle

Four circles are arranged as shown in the figure below. They're numbered from $1$ to $4$. If the diameter of circle $C_2$ is equal to $9$, and $ PT = 6 , QT = 3 \sqrt{5} $. It is also given that $UV ...
2 votes
2 answers
136 views

How to Separate the Equations of Corresponding Straight Lines from a Second-Degree General Equation of Pair of Straight Lines?

I'm trying to understand how to separate the equations of corresponding straight lines from a general second-degree equation representing a pair of straight lines. For example, given the equation: ax^...
0 votes
0 answers
11 views

Why does a strictly affinoid space have a special fiber?

This is a soft and possibly naive question: I am confused by the relation between formal geometry and Berkovich analytic spaces. Roughly speaking, the latter should describe generic fibers of the ...
4 votes
0 answers
158 views

Shapes with simple distance functions.

Given a set $A$ in $\mathbb{R}^2$, the distance function (DF) of $A$ is defined as $$ \delta_A(\mathbf{x}) = \inf\{\|\mathbf{x}-\mathbf{y}\|: \mathbf{y} \in A \} $$ Some sets $A$ have a nice tidy ...
2 votes
1 answer
31 views

Choice of $\lambda,\mu,\nu$ values to determine the parametric equation of the line

Let's consider a Cartesian monometric orthogonal reference system in space $S$. Let's consider the following plane: $$\pi:4x+2y+3z=0$$After choosing a point $P$ not belonging to the plane $\pi$, then: ...
0 votes
1 answer
40 views

Analytic geometry problem, can't get same results solving by middle points & slope?

If the symmetry of point $A(-1, 0)$ on the analytical plane with respect to the line $ax+2y-3=0$ is $B(3, -2)$, what is $a$? Now, when you solve this not by the classical method but by using the ...
0 votes
1 answer
40 views

Determine Intersection Point Quadratic Bezier Curve and Plane

I need to compute the intersection point between a quadratic bezier curve and a plane. Thereby i have to solve the equation for the parameter t. The normal vector is in the dot product to the square ...
46 votes
11 answers
3k views

Dog bone-shaped curve: $|x|^x=|y|^y$

EDITED: Some of the questions are ansered, some aren't. EDITED: In order not to make this post too long, I posted another post which consists of more questions. Let $f$ be (almost) the implicit curve$...
6 votes
3 answers
88 views

Is there a faster way to find max and min of $P = 4x + 2y + 3$ given $|x| + 2|y| = 4$?

Is there a faster way to find max and min of $P = 4x + 2y + 3$ given $|x| + 2|y| = 4$? From a geometrical standpoint, the set of points that satisfy $|x| + 2|y| = 4$ is a diamond on the $Oxy$ plane. ...
1 vote
0 answers
15 views

Morphing between circles

In an algorithm there is one property needed that is a bit unintuitive. For simplicity, we say there is a circle $K_1$ centered at the origin with radius $r_1>0$. Define $$f_1(X):=\frac{x^2+y^2}{...
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0 answers
22 views

Alternative approach: $A, B \in (C): x^2 + y^2 = 1$ and $OAB$ is a right triangle at $O.$ Find min and max of $P= MA \cdot MB,$ with $M(0; 3).$

$A$ and $B$ are two points on the circle centered at $O$ with a radius of $1,$ and $OAB$ is a right triangle at $O.$ Consider $M(0;3).$ Find the min and max of $P = MA \cdot MB.$ I needed to find ...
0 votes
0 answers
32 views

Chords $\overline{AB}$ and $\overline{CD}$ of a circle meet at the point $E$ outside the circle. Prove that

(a) $\angle A\cong\angle C$ (b) $\angle1\cong\angle 2$ (c) $\triangle ADE$ and $\triangle CBE$ are equiangular. I tried upto some extent but did not know I to solve this problem. I know that length ...
25 votes
6 answers
3k views

The formal definition of “angle”

My main question is about the definition of "angle". Many linear algebra textbooks define the angle between two vectors in terms of their inner product. I superficially understand that this ...
0 votes
1 answer
1k views

Finding cartesian equation for a plane in $\mathbb R^4$

Let's assume that I have previously found an orthonormal basis for the plane (dim 2) whose cartesian equation I want to find. Is it as simple as using Gram-Schmidt a third time to find a vector that's ...
1 vote
2 answers
81 views

To derive the equation of the director circle of a standard ellipse using the tangent lines at any two points on the ellipse

I am currently studying conic sections and have encountered the concept of the director circle (or orthoptic circle) of an ellipse"1". The director circle is defined as the locus of points ...
-2 votes
1 answer
49 views

Finding the equations of the lines tangent to $Ax^2+2Bxy+Cy^2+2Dx+2Ey+F=0$ and parallel to $Gx+Hy+I=0$ [closed]

A curve $Ax^2+2Bxy+Cy^2+2Dx+2Ey+F=0$ and a line $Gx+Hy+I=0$ are given. How do I find the equation of the tangent (tangents) to a given curve that is (are) parallel to a given line?
1 vote
0 answers
82 views

Find equation of circles which are orthogonal to a system of coaxial circle.

I am doing an exercise on system of circles: Find the equations of the circles which are orthogonal to all the circles of the system $x^2+y^2+2ax+2by-2\lambda(ax-by)=0\tag*{}$ where $\lambda$ is a ...
-1 votes
0 answers
34 views

Intersection of planes in a 3d space WITHOUT vectors [closed]

We are learning about finding the intersection of planes / lines in 3D space by drawing diagrams, and I don't get it at all. The problem is we aren't doing anything with vectors or graphical diagrams, ...
1 vote
1 answer
62 views

Ruler and compass construction of an inscribed quadrilateral

Suppose that I have a (convex) quadrilateral $ABCD$. In the interior of it I have 4 distinct points $P,Q,R,S$ in general position (i.e. are vertex of a quadrilateral). The question how to construct a (...
1 vote
0 answers
24 views

Proving the Basis of a Scaled Vector Set J

If $ E = \left\{\overrightarrow{e_1}, \overrightarrow{e_2}, \overrightarrow{e_3}\right\} $ is a basis, prove that $ F = \left\{\alpha \overrightarrow{e_1}, \beta \overrightarrow{e_2}, \gamma \...
0 votes
0 answers
14 views

Locus of center of hyperbola that is tangent to coordinate axes

Given the generic hyperbola $ \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 $ which is centered at the origin, suppose we shift it and rotate it, such that one of its branches becomes tangent to the ...
0 votes
1 answer
22 views

Proving Linear Dependence from Cross Product Sum J [duplicate]

Prove that if $\vec{u} \times \vec{v} + \vec{v} \times \vec{w} + \vec{w} \times \vec{u} = \overrightarrow{0}$, then $\{\vec{u}, \vec{v}, \vec{w}\}$ are linearly dependent. Solution: Consider the cross ...
0 votes
1 answer
65 views

Find the inellipse of a triangle with its major axis pointing in a given direction and with a given ratio of major to minor

In a recent post I've investigated the ellipse tangent to the $x$ and $y$ axes as well as a third given line. In this problem, you're given a triangle with known vertices, and your task is to find the ...
0 votes
0 answers
25 views

Regular octagon inscribed in an ellipse

Problem A regular octagon inscribed in an ellipse. The foci of the ellipse $F_1$ and $F_2$ are on the midpoint of a side of a regular octagon. Proof the eccentricity of the ellipse is $e=\frac{\sqrt{...
-4 votes
1 answer
44 views

A surface passing through two different surfaces [closed]

Suppose I have two surfaces $f_1=k_1$ and $f_2=k_2$ in 3D. Then, how do I find the equation of a surface passing through (intersecting) the two surfaces $f_1,f_2$? Like, does $f_1-f_2=k_3$ help? But, ...
0 votes
1 answer
45 views

Line through $A$ parallel to $\alpha$ and incident on $r$.

I tried to solve point $(1.)$ of this exercise, but it seems that no such line exists. I would be very happy if someone could check my work, because it seems strange to me that there isn't a solution....
0 votes
0 answers
18 views

Finding the z coordinate of the center of a sphere when touching a triangle in space without iteration. [closed]

I have the coordinates of three points of a triangle in space. I also have x, y coordinates - the center of the sphere. Is it possible to find the z coordinate of the center of the sphere such that ...
1 vote
2 answers
2k views

The limit about the line connecting the intersection of a circle and the $y$-axis and the intersection of the shrinking circle and a fixed circle

There is a fixed circle $C_1$ with equation $(x - 1)^2 + y^2 = 1$ and a shrinking circle $C_2$ with radius $r$ and center the origin. $P$ is the point $(0, r)$, $Q$ is the upper point of intersection ...
0 votes
3 answers
157 views

Must three distinct spheres always intersect in exactly two points?

Two spheres A, B intersect in a circle, obviously, so a 3rd sphere C intersecting both A and B does so in 2 different circles. It seems to me that the circle of the AC intersection must intersect the ...
2 votes
0 answers
118 views

Find the equation of an ellipse passing through $4$ given points

The purpose of this exercise is to find the equation of the ellipse whose axes are parallel to the coordinate axes, and passing through $4$ given points, where it is assumed that no $3$ of them are ...
2 votes
3 answers
99 views

Find all inscribed ellipses in a given triangle passing through two given internal points

Given a triangle, and two points inside it, I want to determine all the ellipses that are inscribed in the triangle and passing through both of the two given points. My attempt: is outlined in my ...
2 votes
2 answers
82 views

Problem with finding tangent lines to an hyperbola

Given the point $Q(0,6)$ and the hyperbola of equation $$\frac{(x-\frac{1}{2})^2}{a^2}-\frac{(y-1)^2}{b^2} =1$$ $$a=-1+\sqrt{29}/2,b=\sqrt{-2+\sqrt{29}}$$ In the figure it's clear that there are four ...
0 votes
0 answers
16 views

Writing the equation of a family of circles touching two circles

I know that the equation of a circle passing through the intersection of two circles is $S_1+\lambda S_2=0$ and I know that the equation of a circle passing through the intersection points of a line ...
0 votes
0 answers
24 views

Obtaining aggregate rotation and translation of 2D vector field (2d matrix of 2D motion vectors) produced from optical flow

I have the result of a dense optical flow in the form of 2D matrix of 2D motion vectors. I need to isolate the rotation and translation components of the motion to estimate camera rotation and ...
1 vote
1 answer
1k views

A foot of the normal from the point $(4,3)$ to a circle is $(2,1)$ and a diameter of the circle has the equation $2x-y-2=0$.

A foot of the normal from the point $(4,3)$ to a circle is $(2,1)$ and a diameter of the circle has the equation $2x-y-2=0$.Then find the equation of the circle. For the equation of the circle,we ...
1 vote
1 answer
65 views

What is the equation of diameter of a rectangular hyperbola?

So I was doing this question The locus of middle points of parallel chords of hyperbola :$xy=c^2$ The locus of mid-points of parallel chords of a conic is called a diameter or so I was told. I ...
0 votes
1 answer
38 views

Distance to geometric median

The geometric median of a triangle is the point that minimizes the sum of distances to the triangle vertices. Is it true that for any triangle $ABC$ with geometric median $D$, and any other point $A'$,...
1 vote
1 answer
29 views

Geometric Interpretation of Linear Independence and Cross Product

Prove that if $\vec{u}$ and $\vec{v}$ are linearly independent, and $\vec{w} \times \vec{u}=\vec{w} \times \vec{v}=\vec{0}$ then $\vec{w}=\vec{0}$. Interpret geometrically. Solution: $$\vec w\in \left ...
5 votes
3 answers
1k views

Deriving the barycentric coordinates of a triangle's orthocenter, using the areal definition of such coordinates

Wikipedia's "Altitude (triangle)" entry describes the barycentric coordinates of $\triangle ABC$'s orthocenter as $$(\tan A : \tan B : \tan C)$$ How would you prove this using solely the areal ...
10 votes
4 answers
50k views

How can you construct as many intersections as possible with n lines?

If you have $n$ lines, it seems to be obvious that you can have at most $\frac{n^2-n}{2}$ intersections: $n = 1$: Obviously you need two lines to intersect, so the maximum number of intersections is ...
-2 votes
0 answers
45 views

General proof of this special triangle property

Consider any general triangle $ABC$ contained in a two dimensional plane. Three circles are drawn having the sides of triangle $ABC$ as diameters. Prove that the centre of the circle, which is ...
-1 votes
0 answers
21 views

Determine MPH of a Sphere From Diameter Change Using Image Processing

I'm looking for a formula to calculate the speed of a ball coming directly at or away from you solely based on the the change in diameter over time. For example, if I video recorded a ball that was a ...
0 votes
1 answer
32 views

An ellipse $\mathcal{E}$ touches a fixed ellipse $\mathcal{C}$ at $A$, prove the length of semi-major axis of $\mathcal{E}$ is constant

$\mathcal{C}$ is an ellipse with center $O$ and semi-major axis length $=a$, semi-minor axis length $|OB|=b$. $A$ is a point moving on $\mathcal{C}$. $E$ is a point on $\mathcal{C}$ such that $OE$ is ...
1 vote
1 answer
43 views

Number of circular cylinders/cones through an ellipse

Given an ellipse $x^2 / a^2 + y^2 / b^2 = 1\;(a>b>0)$ on $xy$-plane. Then an arbitrary elliptical cylinder through the ellipse has equation $(x / a + u z)^2 + (y / b + v z)^2 = 1$ for some $u,v\...
2 votes
0 answers
33 views

$(C): (x-4)^2 + (y-3)^2 = 4, A(-1;0), B(-3;0), M \in (C).$ How to find min/max of $P = MA + MB?$

$(C): (x-4)^2 + (y-3)^2 = 4$ $A(-1;0)$ $B(-3;0)$ $K(-2;0)$ $M$ is a point on the circle. I wonder what the method is to find the minimum value of $MA + MB$ as $M$ is moving on $(C)$. I'd love to hear ...
3 votes
1 answer
212 views

Prove that every root of $P(z)$ in the closed unit disc has multiplicity at most $c \cdot \sqrt{n}$, where $c = c(M) > 0$ is constant depending on $M$

Problem statement: Let $P(z)$ be a polynomial of degree $n$ with complex coefficients, $P(0) = 1$, and $|P(z)| \leq M$ for $|z| \leq 1$. Prove that every root of $P(z)$ in the closed unit disc has ...
4 votes
0 answers
139 views

Why is $ \mathfrak{Mod}(A_{Y}/f) $ a thick subcategory of $\mathfrak{Mod}(A_{Y})$?

Let $f: Y \to X$ be a continuous map and $\mathfrak{Mod}(A_{Y}/f)$ be the full subcategory of $\mathfrak{Mod}(A_{Y})$ (categories of $A_{Y}$-modules, with $A$ a fixed ring) whose sheaves $\mathcal{F}$ ...
2 votes
2 answers
2k views

Check if the point lies within the 3d volume

What is the condition to check if any point $(x,y,z)$ lies within any 3D shape (cuboid, tetrahedrons etc)? Is there any generic condition that I can use to compute it for any shape?

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