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

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Range of slope of line PQ

Let $A(-1,0),B(3,0)$ and PQ be any line passing through (4,1).The range of the slope of PQ for which there are two points on PQ at which AB subtends a right angle is $(\lambda_1,\lambda_2)$,then what ...
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1answer
21 views

Sorting triangles by hypotenuse length

I have some points in $xy$ space and I need to sort distances between these points. If I calculate real distance, then I need to perform $\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$ and this is very time ...
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1answer
53 views

Intersection of Three Planes proof

I'm supposed to be making a study guide answer for this question, but I'm struggling with proof. Show that the three planes intersect at the point provided that Note that the ...
3
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3answers
36 views

Find the equation of a circle that intersects the $y$-axis at the origin and at the point $(0,6)$ and also touches the $x$ axis. - basic question

Find the equation of a circle that intersects the $y$-axis at the origin and at the point $(0,6)$ and also touches the $x$ axis. Okay, so I wasn't sure how to do this so I looked at the answer at ...
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1answer
15 views

Angle bisector equation and orthocenter given, vertex to be found

In triangle $ABC$, let $A(3,4)$ and the equation of angle bisector of $B$ is $y=x$. If orthocenter of triangle is $(2,2)$ and $B(h,k)$, then find $(h,k)$. I cant solve this question, I am confused, ...
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Question based of orthocenter distance from angular points

In an acute angled triangle ABC,$\angle A=20^\circ $,let D,E,F be the feet of altitudes through A,B,C respectively and H is the orthocenter of $\bigtriangleup ABC $.Find ...
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1answer
27 views

Polarity on a Hyperboloid of one sheet

Given a quadric $Q = \{v \in \mathbb{R}^n \mid \alpha(v,v) = 1\} \subset \mathbb{R}^n$, defined by a bilinear form $\alpha: \mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}^n$, and an affine subspace ...
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1answer
40 views

Equation by Graph

Given a random Graph,is there any known way to find an equation for it ? If I create a random graph is there a way that i can find an equation that totally describes my random graph?
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2answers
34 views

Angle bisector related question

The internal bisectors of the angles of a triangle ABC meet the sides in D,E,and F.Show that area of the triangle DEF is equal to $\frac{2\Delta\times abc}{(b+c)(c+a)(a+b)}$,here $\Delta $is area of ...
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2answers
57 views

Finding the equation of a circle.

A circle of radius $2$ lies in the first quadrant touching both axis. Find the equation of the circle centered at $(6,5)$ and touching the above circle externally. Let me share how I answered this ...
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5answers
73 views

Find the equation of the circle.

Find the equation of the circle whose radius is $5$ which touches the circle $x^2 + y^2 - 2x -4y - 20 = 0$ externally at the point $(5,5)$
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3answers
23 views

Derivation of the equation for the envelope

Suppose we have a family of curves on the plane. The equation of the curves is given by $$ f(x ,y ;t) = 0 . $$ Here $t$ is the parameter. On Wiki, the equations determining the envelope of this ...
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1answer
14 views

Point of Division of a Line Segment [on hold]

The segment joining the points $(-1, 0)$ and $(-3, -6)$ is extended each way a distance equal to $3/4$ its original length. Find the terminal points.
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1answer
34 views

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
19 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
23 views

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
21 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
42 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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0answers
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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
21 views

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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2answers
44 views

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
27 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
61 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. [closed]

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
37 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
60 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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0answers
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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
31 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
48 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
42 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
32 views

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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0answers
35 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
110 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
14 views

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 ...