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

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equally spaced on circle question

Define $$\|\vec{x}\|:=\sqrt{\alpha^2+\beta^2},$$ where $\vec{x}:=(\alpha,\beta)\in \mathbb{R}^2.$ Set $$\mathbb{S}^1:=\{\vec{x}\in \mathbb{R}^2: \|\vec{x}\|=1\}\quad \quad and\quad \quad ...
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1answer
16 views

$|\vec{r}-\vec{r}_1|=\frac{1}{2}|\vec{r}-\vec{r}_2|$ is the equation of a sphere?

I am told that the set of all $\vec{r}$ for which $|\vec{r}-\vec{r}_1|=\frac{1}{2}|\vec{r}-\vec{r}_2|$ is true forms a sphere---however, my semi-intuitive reading of this equation puts a "weaker ...
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1answer
14 views

Orthocenter and coordinates of a vertex

In a triangle ABC,the vertex A is (1,1) and orthocenter is (2,4).If the sides AB and BC are members of the family of straight lines $ax+by+c=0$,where $a,b,c$ are in arithmatic progression.Then the ...
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0answers
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Disk in analytic topology vs. the spectrum of a Henselian DVR in etale topology

In this informative and concise set of notes on vanishing cycles by Donu Arapura, it is stated that the theory of vanishing cycles ports nicely to the etale world if the role of the disk is replaced ...
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1answer
20 views

Equivalence of euclidean and analytic geometry [on hold]

I read about the axioms of euclidean geometry. How is analytic geometry rigorously defined? What are the axioms? And most important: How to prove that all the results proved in analytic geometry are ...
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2answers
39 views

Minimum value of an an expression

Find the minimum value of $(\alpha-\beta)^2+(\sqrt{2-\alpha^2}-\frac{9}{\beta})^2$ where $ 0<\alpha<\sqrt{2}$ and $\beta>0$ My attempt: In my view,this minimum value is the shortest ...
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2answers
15 views

How do I find intersections between a circumference and an equilateral hyperbola?

Let's say I have a circumference with the equation $x^2 + y^2-10=0$. This circumference has a point A $(1;3)$ which which passes thorough an equilateral hyperbola $xy=3$. I would like to find all the ...
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3D Geometry-If alpha /2 ,beta /2 and gamma /2 are the angles with 3 axes.Then, cos alpha + cos beta + cos gamma [on hold]

If alpha /2 ,beta /2 and gamma /2 are the angles which a line makes with the x,y,z axes respectively.Then, cos alpha + cos beta + cos gamma = ? a-> 1 b-> (-1) c-> 2 d-> 3
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2answers
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Find the equation of an equilateral hyperbola passing through a point of a circumference

Let's say I have a circumference with the following equation $x^2+y^2-10=0$, the coordinates of its center are $(0;0)$ and its radius is $\sqrt{10}$. I need to find the equation a equilateral ...
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1answer
23 views

co-ordinate geometry question 4 [on hold]

Find the equation of the straight line which passes through the point $(3, 2)$ and its gradient is $\frac{3}{4}$ . Find the co-ordinates of the points on the line that are $5$ units away from the ...
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Find a plane with distance $3$ from $3x-y-z = 0$

I need to find a plane such that its distance from the plane $3x-y-z = 0$ is $3$. Since distance is defined only for parallel planes, I already know that they have to be parallel, and then, the ...
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4answers
51 views

Find the distance between the point $(0,0,0)$ and the plane $2x+3y+z=1$ [on hold]

Find the distance between the point $(0,0,0)$ and the plane $2x+3y+z=1$. So I know in order to find the distance I need two points. How do I find a point in the plane?
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2answers
22 views

Find the cartesian equation of the locus of the set of points of $P$ problem.

Find the cartesian equation of the locus of the set of points of $P$. $P$ is at a constant distance of five units from the line $4x-3y=1$ I don't have much intuition on how to solve this one. ...
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1answer
56 views

Point on the Plane, a Triangle, and a Lower Bound of a Ratio Sum

Let $ABC$ be a triangle on the Euclidean plane. At which point $P$ on the plane does the ratio sum $\frac{PA}{BC}+\frac{PB}{CA}+\frac{PC}{AB}$ attain its minimum value? Prove also that, for any ...
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1answer
39 views

Can anyone help me with this contradicting graphs?

While studying SAT MATH 2 , I tried to solve the following problem but faced some difficulty. The problem goes ........ In the graph of the parametric equations $x= t^2+t$ , $y=t^2-t$ A) $x\ge 0$ ...
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1answer
81 views

reference on $\sqrt{ax}+\sqrt{by}=c$ as a parabola?

Does anyone have a reference on the equation $$\sqrt{ax}\,+\sqrt{by}=c\ ?$$ Clearing square roots and rearranging gives $$ax+by = \frac{(ax-by)^2+c^4}{2c^2}$$ This is the equation of a parabola, so ...
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0answers
26 views

Find a line in a plane perpendicular to another given line. [on hold]

Find a line in the plane of $(0,0,0)$, $(2,2,0)$, and $(0,1,-2)$ and is perpendicular to line $x+\dfrac13=y-\dfrac12=2z$. After years of working in an unrelated tech position, I decided to pull out ...
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2answers
35 views

differentiating an integral with respect to a variable which also affects the region of integration

I am considering taking the derivative of the function $$F(\mathbb{x_1},\mathbb{x_2},\mathbb{x_3}) = \displaystyle \int_{V_1} ||x-\mathbb{x_1}||\phi(x)\,dx + \int_{V_2} ||x-\mathbb{x_2}||\phi(x)\,dx ...
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3answers
58 views

Drawing circumference issue

I'm a developer, and I'm developing an app on Google Maps. At the moment, I'm trying to draw a circle on the map. For getting all the points I need, I'm using the following formula: \begin{equation} ...
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curve paramter doubt [closed]

what is segment number and paramter value in curve .. what does it means .. pealse let me know ... ULong m_SegNum; //Segment number double m_ParmVal; //Parameter value
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Provide solution to geometrical problem [closed]

Given points belonging to great circles (within an error: ±0.5⁰show): A- Which belong to the same circle? B- Which circles intersect? C- Prove the points of intersection are unique. With the data ...
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1answer
30 views

Interpolating Random Points

I have a list of (x,y) co-ordinates that need to be interpolated. The co-ordinates are not necessarily part of a function. Therefore, polynomial interpolation will not work. Is there a way to use some ...
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1answer
47 views

Rotation of complex numbers in a complex plane. Check my work?

Say that $c_1 = -i$ and $c_2 = 3$. For this problem, let $z_0$ be an arbitrary complex number. We can rotate $z_0$ around $c_1$ by $\pi/4$ counterclockwise to get $z_1$. Next, we canrotate $z_1$ ...
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4answers
118 views

how to prove that the circle $(x-a)^2+(y-b)^2=a^2+b^2$ is passing through point $(0,0)$

How can one prove that the circle $(x-a)^2+(y-b)^2=a^2+b^2$ is passing through point $(0,0)$? I know that if i put: $x=y=0$, i will get: $(0-a)^2+(0-b)=a^2+b^2=a^2+b^2$ But that's not a proof but ...
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4answers
105 views

Median of triangle

I know that a median of a triangle is a line joining one of the vertices to the mid-point of the opposite side. For example, in a triangle OAB, O is the origin, $A$ is the point $(0,6)$ and $B$ is ...
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2answers
44 views

Find value of $t$ between the difference of 3D vectors.

Hint: The distance between $2$ vectors equals the magnitude of their difference. What is the value of $t$ for which the vector $\mathbf v = \begin{pmatrix} 2 \\ -3 \\ -3 \end{pmatrix} + ...
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2answers
61 views

Distance involving 3D lines and vectors.

In this problem, a = \begin{pmatrix} 5 \\ -3 \\ -4 \end{pmatrix} and b = \begin{pmatrix} -11 \\ 1 \\ 28 \end{pmatrix} Vectors p and d exist such that the line containing a and b can be expressed in ...
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1answer
41 views

Determining positions of straight line

I need to find out the relative positions to each other a straight line, first I'm trying to check if they are coplanar but I get an unknown variable. Can anyone help me on how to solve this part of ...
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2answers
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3D line in a 3D plane. Find the intersection of the two.

(I'm new to Math.StackExchange, so if you see any errors, please comment below!) $\mathcal{P}$ is the plane containing the three points $(-3,4,-2)$, $(1,4,0)$, and $(3,2,-1)$. $\ell$ is the line ...
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0answers
28 views

Lines on a specific cubic surface

Consider the cubic surface given in affine coordinates by the equation $x^2+y^2=g_3(z)$ ($g$ is a polynomial of degree 3 s.t. the cubic is smooth). Is it possible to write down explicitly the ...
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2answers
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Find a bisector point of a circle

The coordinates of $A=(x_{0},y_{0}$) and $B=(x_{1},y_{1}$) are given. How to find the coordinates of $C$ and $D$ as per given information below. ABC is equilateral triangle such that $AB=BC=CA$ ...
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34 views

What's the area of the shape defined by all points whose distances from two focal points multiply to give the same product?

This shape, which I call the multiplicoid, is the equivalent of, and very similar to, an ellipse. However, instead of the distance between each point and the two focal points summing to a constant, ...
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4answers
108 views

Is it possible to put an equilateral triangle onto a square grid so that all the vertices are in corners?

In the following collection of problems - arXiv:1110.1556v2 [math.HO] - the following question is posed: Is it possible to put an equilateral triangle onto a square grid so that all the vertices ...
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2answers
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Centering Text on Image after padding values

I'm creating an Image, the Image has text on it. The text I want it to be center on the Image but the Image has padding values that I assigned. The equation I'm using and the values for the ...
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1answer
21 views

Coordinates of regular octagon

There are coordinates of two vertices in a regular octagon $$\{A_0,A_1,...,A_7\}$$ with Cartesian coordinates $A_0 = (2,-4)$, $A_2 = (0,0)$. The task is to find coordinates of all other vertices. ...
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1answer
44 views

Find a plane that passes through a point $p$ but is perpendicular to a line $\ell$ [closed]

The point $p$ that I am using is $(1,6,-4)$ and line $\ell$ is $$ \begin{cases} x = 1 + 2t\\ y = 2 - 3t \\ z = 3 - t \end{cases} $$ Other points for the line I have found are $(1,2,3)$ and ...
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3answers
65 views

is there an online tool for solving equation of a line? [closed]

I have as input two points. But the input points contain variables (constant references). I've seen some tools online but they require numeric values. The two given points are specified by constant ...
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2answers
46 views

How can one calculate the limit of $\frac{1}{x^2-9}$ as x approaches -3 and 3 by hand? [closed]

Reviewing math for college after a gap year and so I know this is probably a pretty elementary question, but let me know if it has any interesting implications or alternative solutions or if it ...
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1answer
41 views

Intersection of two lines in complex numbers given four points [closed]

How to find the point of intersection of two lines, given four points, two of which are on each line, in complex numbers? Thank you!
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An integral from the integral geometry about the isoperimetric inequality.

The problem is from the book "Integral Geometry and Geometric Probability" by Santalo (1976), Chapter 1.3.5, Notes and Exercises (page 37). Given a convex closed curve $C$. Let $A_1$, $A_2$ be the ...
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1answer
23 views

Strange “form” of the set of vertices $C(x,y,z)$ such that $ABC$ is a right triangle with hypotenuse $AB$

Let $A(1,-3,4)$ and $B(3,-2,-1)$ and find the set of all $C(x,y,z)$ such that $ABC$ is a right triangle with hypotenuse $AB$ What I did $$AB=(2,1,5)$$ $$BC=(x-3,y+2,z+1)$$ $$AC=(x-1,y+3,z-4)$$ ...
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4answers
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Integrating $\sqrt{1-x^2}$ without using trigonometry

I am a beginning calculus student. Tonight I had a thought. Maybe I could calculate $\pi$ using integration, but no trig. The problem is that I don't really know where to start. I thought perhaps I ...
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2answers
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Given the endpoints of a line segment, develop the equation of its perpendicular-bisector

Find the equation of the perpendicular bisector of $AB$ for: $A(1\mid 3)$ and $B(-3\mid 5)$. What I did: $m=\frac{3-5}{1+3}=-\frac12$ for the slope of $AB$ $(\frac{3+5}2\mid\frac{1-3}2)=(4\mid -1)$ ...
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2answers
42 views

Deduce the inequalities $3\lt \pi \lt 12(2-\sqrt{3})$, by calculating the areas of regular twelve-sided polygons.

Calculate the areas of regular dodecagons (twelve-sided polygons) inscribed and circumscribed about a unit circular disk and thereby deduce the inequalities $3\lt \pi \lt 12(2-\sqrt{3})$. This is a ...
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A better way to answer this question

So my team and i were asked this question a few years ago on a small Math-A-Thon on my hometown. It went something like this: "We need to transport a neon tube (or any tube, who cares) of 92cm ...
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2answers
44 views

Find the Ratio $BM \colon ME$

In Triangle $\Delta ABC$, the Point $D$ is on $BC$ such that $D$ divides $B$ and $C$ in the Ratio $1 \colon 3$ and there is a point $E$ on $CA$ such that $E$ divides $C$ and $A$ in ratio $1 \colon 3$. ...
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1answer
28 views

Equation of the locus

Find the equation of the locus of a point $P = (x, y)$ when the sum of the squares of the distances from $P$ to the points $(a, 0)$ and $(-a, 0)$ is $4b^2$, where $b \geq \dfrac{a}{\sqrt{2}}$?
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1answer
43 views

How to get rid of the term with $xy$?

I'm trying to put this conic on an identifiable form. $$4x^2-4xy+y^2+20x+40y=0$$ I know that the term $xy$ implies that I need to rotate the conic so that $xy$ vanishes. But I've read on some books ...
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22 views

Should the expanded expression of a quadratic form be equals to It's original expression?

Sorry if the question is a little misleading, but I have no better way to express it. The text below should clarify. Suppose I have the equation of a conic: $x^2+y^2+z^2-2x+3y+z+2=0$, with this I ...
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1answer
13 views

Point within a Cube in 3D environment

I have a cube in 3D space with 8 corner points with their X,Y,Z Coordinates. I know how to test if any given point lies inside a cube by Comparing their coordinates to be greater or smaller than the ...