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

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3
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1answer
45 views

Area enclosed between half lines in polar space

I don't know if the anwser to my question is obvious because I cannot find any explanation anywhere on google. Question The blue region $R$ is bounded by the curve C with equation $r^{2} = ...
0
votes
2answers
43 views

Length of a line in an isosceles triangle. (mind boggling )

In an isosceles triangle ABC, side AB and AC are equal in length. There exists a point D on the side AB. The angle BAC is theeta . The side AD is two units smaller than AC .What is the generalized ...
26
votes
3answers
1k views

Fascinating Lampshade Geometry

Today, I encountered a rather fascinating problem in a waiting room, which is embodied in the image below. Notice how the light is being cast on the wall? There is a curve that defines the ...
0
votes
1answer
27 views

To find an intersection point between two planes with only the direction vector

Find the intersection between two planes $x−3y−2z = 2$ and $2x+y+3z = 1$ Solution: $(1)$$\quad n_1 \times n_2 =\langle −7,7,7\rangle =7 \langle −1,1,1\rangle$. $(2)$ To find one intersection ...
2
votes
0answers
20 views

finite morphism (algebraic) vs finite morphism (analytic)

Let $X$ and $Y$ be two algebraic varieties (reduced schemes of finite type) over $\mathbb{C}$. Let $f : X \to Y$ be a morphism of schemes. Let $X^{an}$, $Y^{an}$ and $f^{an}$ the corresponding ...
0
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1answer
49 views

How to find the midpoint given the linear equation

Given the linear equation: $$y = 7247.5188 -2395.0376x$$ how do I find the midpoint of this line?
2
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0answers
32 views

Locus of centre of circle in Lambert theorem

A beautiful theorem, when three tangents to a parabola form a triangle,the focus of the parabola lies on the circumcircle of the triangle. But what is the locus of the centre of the circumcircle of ...
3
votes
0answers
47 views

Is there an algebraic description of the ring of analytic functions on the real projective line?

Apologies for the long question. Let $X=\mathbf P^1(\mathbf R) \subseteq \mathbf P^1(\mathbf C)$ be the real projective line. Let $\mathcal O_X$ be the sheaf of real-analytic complex-valued functions ...
2
votes
1answer
25 views

Vectors triangle problem.

Let D be the midpoint of the side BC of the triangle ABC Verify that: $$\vec{AD}=\tfrac12\big(\vec{AB}+\vec{AC}\big)$$
1
vote
2answers
48 views

how to calculate the angle between the tangents of the curve?

$y=(-3/2)x$ and $y=(-2/5)x$ intersect the curve $$3x^2+4xy+5y^2-4=0$$ at points $P$ and $Q$ .find the angle between tangents drawn to curve at $P$ and $Q$ .I know a very long method of finding ...
4
votes
0answers
69 views

Tangent developable of helix.

Let $T$ be union of tangent lines to helix $C=(\cos x, \sin x,x)$. 1) I want to prove that $T - C$ is a smooth manifold and find equation for $T$. 2) I want to find how many times a line can ...
1
vote
1answer
48 views

Find all points with whole- number's coordinates inside the area of polygon

I've got the polygon with n angles. I know the coordinates of its apexes (their coordinates are integers), but I don't know the total area of that polygon. Is there any way to count how many points ...
0
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0answers
38 views

Computer program for visualizing multivariable Calculus topics

I am an undergraduate studying multivariable Calculus. However I have difficulty visualizing concepts. In single variable calculus I can visualize stuff, for example when one talks about derivatives, ...
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vote
0answers
42 views

building a polytop from polytop and finding its volume

Let $P$ be a symmetric polytope with $M$ vertices. Suppose we subdivide this polytope into $M$ equal parts $A_i, i=1, \ldots, M$ such that each part $A_i$ correspond to one vertex, $v_i, i=1, \ldots, ...
0
votes
1answer
17 views

About a/the definition of plane.

Let $P$ be a point in 3-space and consider a located vector $ \overrightarrow {0N}$. We define the plane passing through $P$ perpendicular to $ \overrightarrow {0N}$ to be the collection of all ...
0
votes
1answer
49 views

Shortest distance between parallel line and plane

I've been doing questions regarding the shortest distance between lines/planes and points , and I've come across a question asking to find the shortest distance between a line and a plane which are ...
3
votes
0answers
35 views

Applications of the quartic curve $x^2y^2-1=0$?

The quartic curve $x^2y^2-1=0$ is equivalent to the union of the hyperbolas $xy-1=0$ and $xy+1=0$, i.e., it's a rectangular hypobola superimposed with a copy of itself rotated by 90 degrees. Does this ...
1
vote
1answer
43 views

Calculating tangent on ellipse

I want to calculate the slope of the tangent at one point of an ellipse whose centre is shifted towards the coordinates $(x_c;y_c)$ and also rotated by an angle $\alpha$ around its centre. Now, I have ...
1
vote
4answers
77 views

A triangle has to find its third side.

Problem: (Euclid had a triangle in mind - I am including this line so that future googles come across this question) The triangles longest side is $20$ and another side is $10$. Its area is $80$. ...
1
vote
1answer
39 views

Crazy rectangles, semi-circles, and circles!

Problem is to find the ratio of the area of the circle to that of the semi-circle. Note that points $F$ and $E$ weren't given in the original diagram, and that the circle at the top-right ...
0
votes
1answer
18 views

Proving congruency of triangles

Question: Given $AB$ is diameter, $C$ and $D$ lie on circumference, $AB = 15cm$, $AC = 12cm$, $BD = 9cm$, find area of quadrilateral ABCD. Note that the points $O$ and $Q$ were not in the ...
1
vote
2answers
41 views

What is a homographic solution in three body problem?

I came across Saari's homographic conjecture in Three Body problem. I need more information on what exactly is a homographic solution and how is it different from a homothetic solution?
0
votes
1answer
23 views

Locus of the centre of a circle $\Gamma$

Let $\Gamma_1,\Gamma_2$ be two circles centred at the points $(a,0),(b,0);0<a<b$ and having radii $a,b$ respectively.Let $\Gamma$ be the circle touching $\Gamma_1$ externally and $\Gamma_2$ ...
0
votes
1answer
19 views

How to prove that given a line L prove that all points of a fixed distance k form two lines parallel to L

How can I prove start this?I know intuitively since they never meet they are parallel, but I don't think that is a direct proof.
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0answers
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Is there a Focal Point/Area/Line of a Parabola for not perpendicular Lines

I'm not sure if this is mathematical enough for this forum, since it's my first post, but please don't be too harsh! So my question is: If the incoming lines of a Parabola come in perpendicular to ...
1
vote
1answer
57 views

Conic Sections and Complex numbers

If $\omega$ is a complex number such that |$\omega$| does not equal 1, then the complex number $$z = \omega + \frac{1}{\omega}$$ describes a conic. The distance between the foci of the conic described ...
1
vote
1answer
30 views

Geometric Intuition Of Partial Derivative in 2D space

$$f(x,y)=3x^2+5y^2-2x+3y+7=0\\f'x=6x-2=0\\x=\frac {1}{3}\\f'y=10y+3=0\\ y=\frac {-3}{10} \\ O(\frac {1}{3},\frac {-3}{10})$$ As you see, we've used the partial derivative to find the center point of ...
0
votes
1answer
23 views

Permutation Combination

Let $C = \{(i, j)|i, j \in \mathbb Z,\; 0 ≤ i, j ≤ 24\}$. How many squares can be formed in the plane all of whose vertices are in $C$ and whose sides are parallel to the X−axis and Y − axis?
0
votes
1answer
32 views

Angle bisectors for pair of lines

Suppose you have a pair of lines passing through origin, ax^2 + 2hxy +by^2 = 0, how would you find the equation of pair of angle bisectors for this pair of lines. I can do this for 2 separate lines, ...
0
votes
1answer
14 views

Coplanar vectors

Prove that if $$\vec{a}\times\vec{b}+\vec{b}\times\vec{c}+\vec{c}\times\vec{a}=0$$ then $\vec{a},\vec{b},\vec{c} $ are coplanars. One thing I know is that i have to get ...
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0answers
47 views

Family of circles touching a line

I found this in a book but I am not able to understand how they got this result. It goes the equation family of circles touching a given line $(y-y_1)=m(x-x_1)$ at $(x_1,y_1)$ for any value of $m$ is ...
1
vote
1answer
49 views

Finding a point on a circle that has a distance L (arc length) from another point

Given the coordinates of a single point on a circle and a length of an arc $L$, how do I find the coordinates of another point? Or, to put in another form: I have the radius $r$, the length of the ...
4
votes
2answers
89 views

Applications of Stein spaces in Algebraic Geometry

I want to know where are essential applications of the theory of Stein spaces in algebraic geometry. I heard Cartan's theorem A & B were used in Serre's GAGA, but are there any other applications? ...
1
vote
2answers
59 views

Finding an equation of a circle with a given center and a tangent line.

My math homework is finding an equation of the circle. Given that the center is at (-3,-5) and tangent to the line 12x + 5y =4. I don't know how to solve this since our professor didn't teach this to ...
0
votes
0answers
15 views

Scalating the derivative, and rotating about some angle.

If the angle between a straight line $y=ax$ w.r.t to the x-axis is alpha, then we know that the derivative $a = \tan(\alpha)$. Then I'm interesting in the angle we are rotating alpha about the ...
4
votes
0answers
62 views

Soft: Why does the existence of a singularity cause problems for deRham cohmology?

I've heard that if a variety has a singularity then the deRham theory has "problems". What exactly are these? Im guessing there is some sort of issue with the defintion of a differential form, but ...
0
votes
2answers
77 views

How to justify the similarity of objects in mathematics form

I have developed a system to trace the outlines of (images of) objects. Now I want to test whether two independent traces represent a common feature. Imagine two people (or machines) tracing the ...
0
votes
0answers
29 views

dimension of tangent space to a boundary point of a convex shape

I have a basic question regarding the dimension of the tangent space at a point $P\neq0$ that lies on the boundary of a pointed convex cone with its point centered at 0. For a 3D cone that is ...
2
votes
2answers
31 views

Prove $(|OP|)+ |PQ|)^2 > |OQ|^2$

I did all the algebra and for some reason I'm getting 0 > $y_2^2$ which is clearly wrong. Where did I mess up at?
0
votes
1answer
25 views

Equation of line passing through a point parallel to a given line

I have the point $(2,-5)$ and an equation $y-4 = 2x$ which is a straight line. I want to make another equation from the $(2,-5)$ that is parallel to $y-4 = 2x$ and you can only do this by making the ...
0
votes
2answers
51 views

Hyperbolas - Standard Form

This is probably a simple question but if $y = \frac{1}{x}$ is a hyperbola, then how does it comply with the standard form of a hyperbola?
2
votes
1answer
30 views

Given two closed curves, when is their minkowski sum differentiable?

Suppose you are given closed curves, $\gamma_1$ and $\gamma_2$, which define convex figures in the plane. If we take the minkowski sum of $\gamma_1$ and $\gamma_2$, when is the resulting curve ...
4
votes
2answers
19 views

assessing linear relationships as logarithms

I am teaching myself maths. I am not sure how to approach this problem. It is assessing linear relationships of the form $y=mx+c$ as logarithms. Here I have gotten as far as taking the gradient ...
2
votes
3answers
65 views

Equilateral triangle inscribed in a ellipse

"Given any point on a ellipse, is it always possible to inscribe an equilateral triangle, with a vertex coincident with that point, in the ellipse?" I thought I could use analytical geometry, but ...
0
votes
1answer
45 views

Geometric interpretation of a complex solution

A straight line in 2-D $x+y=3$ and a circle in 2-D $x^2+y^2=4$ do not have a point of intersection in the plane containing the two. But on solving these equations analytically, on gets 2 complex ...
0
votes
1answer
35 views

Equation of horizontal/vertical line and changing to $y=mx+c$ format

I've been given the equation $2x-3y=5$. I was wondering whether this is a horizontal or vertical equation and how would I rearrange this to $y=mx+c$. I know that this is a fairly basic equation but ...
4
votes
1answer
119 views

Tensor notation (practicing)

I'm praticing tensor notation, and I want to prove this way that given vectors $A,B,C,D$ then $(A \times B) \times (C \times D) = \det(A,C,D)B - \det(B,C,D)A$, where $\det$ means the triple product. ...
0
votes
1answer
60 views

List of topics for basic calculus (1st,2nd,3rd semester)

I am an computer science student, currently studying in 2nd semester. Therefore my math courses are pretty weak. Although I "aced" them, I still feel I could use some extra basic calculus knowledge in ...
2
votes
3answers
164 views

Analytic Geometry

How does one solve: Find the equation of the circle which has it's center on the line $y= 3-x$ , and which has as tangents the lines $ 2y-x = 22, $ $ 2x+y=11 $ ?
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0answers
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Circle Tangent question

I would like to ask for assiatance on the following: Find the eqation of a circle, with a radius of$\sqrt 2$ , which also has as tangetns the lines: $ y=x+2 $ , $ y=-7x $. It is known that the ...