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

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2answers
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Existence of n-dimensional polyhedron given edges

The following assertion is true in $2$ and $3$ dimensions: Given $\sigma_{ij},\ 1\leq i\neq j\leq n$ with $\sigma_{ij}=\sigma_{ji}$ and $\sigma_{ij} \leq \sigma_{ik}+\sigma_{kj}$, then there exist ...
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2answers
2k views

Three-circle intersection for circles of unbounded integer radius

I have three circles. One is at $(0,0)$ and has radius $n$, another has is at $(1,0)$ and has a radius $m$, and the third is at $(0.5, \sqrt{0.75}))$ and has a radius of $o$. All of the radius values ...
3
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4answers
4k views

How to calculate the two tangent points to a circle with radius R from two lines given by three points

I need to calculate the two tangent points of a circle with the radius $r$ and two lines given by three points $Q(x_0,y_0)$, $P(x_1,y_1)$ and $R(x_2,y_2)$. Sketch would explain the problem more. I ...
2
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2answers
937 views

Problem Solving Question Relating to Directions and finding Burger Jack

I stopped at a street corner and asked for directions to Burger Jack. Unfortunately, the person I wasked was Larry Longway, whose directions are guaranteed to be too complicated. He said,"You are now ...
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2answers
1k views

Perpendicular Vectors

Find the equation of the line passing through a point $B$, with position vector $ \vec b$ relative to an origin $O$, which is perpendicular to and intersects the line $\vec r= a+ \lambda \cdot c$, ...
4
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2answers
641 views

Applied ODEs in trajectory problem

I'm having a hard time solving this problem: Let there be a town $A$ in a shore of a river. Let $x=0$ be the shore. Let $(0,0)$ be the location of the town. Let $B$ be another town, in the ...
3
votes
1answer
587 views

finding one circles radius so that it tangentially touches two other set circles

I am designing a water fountain on google sketchup and have run into a problem. I am designing the contours of the stone in the fountain. I would attach a picture of the problem but i need 10 ...
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3answers
186 views

Finding the $Y$-intercept of a line, given two points

I'm not sure how best to ask this, so I'll try to explain. Say I have a line drawn between the points $(-1,50)$ and $(2,30)$. How can I figure out the $Y$-value when the line crosses the $X$-value of ...
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0answers
75 views

Relationship between Number of circles required to surround a circles and the distance function?

In Why is a circle in a plane surrounded by 6 other circles, the implicit assupmtion is the distance is Euclidean, my question is: Are there any relation between the distance function being used and ...
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1answer
2k views

how can I obtain enclosed area between two circles in cartesian coordinates?

In the diagram below (from here fig.2, page.5) the enclosed area between two circles (shaded area) has been indicated $a_{t+\delta_{t}}$. Can anyone help me how can I compute this? is it true? ...
3
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1answer
131 views

What volume does $2x \le x^2+y^2+z^2 \le 4x$ represent?

I'm evaluating $\iiint_V f(x,y,z) dV$ where V is defined by $$2x \le x^2+y^2+z^2 \le 4x $$ To simplify things I swapped x and z, and moved to spherical coordinates: $$ 0 \le \theta \le 2\pi, 2 \cos ...
5
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1answer
125 views

An equivalence relation on regions of the plane.

Let $R\subseteq\mathbb{R}^2$. Consider the set of all "horizontal sections" $H_R =${$Rb|b\in\mathbb{R}$}, where $Rb=${$a\in\mathbb{R} | (a,b)\in R$}. Similarly consider the set of "vertical sections" ...
4
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1answer
157 views

How can one use the logarithm function to define angles?

In dealing with the complex logarithm function, I read that the imaginary part of $\log w$, is also called the argument of $w$, $\operatorname{arg }w$, and it is interpreted geometrically as the angle ...
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4answers
724 views

Finding the equation of a circle

$A=(3,1)$ and $B=(-1,-1)$ are points on a circle of center $(k, -3k)$ find the value of $k$ I begin by assinging the values $\ g = -k $ and $\ f=3k $. I then substitute $(3, 1)$ and $ g= -k, f= ...
6
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1answer
367 views

Why is $m$ used to denote slope?

What is the reason, historically, that the letter $m$ is used to denote the slope of a line?
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0answers
44 views

degeneracy loci of dimension $2$

Let $X$ be a smooth complex projective variety of dimension $n \ge 4$ and let $F$ and $E$ be two (holomorphic) vector bundles of rank $f$ and $e$ over $X$. Given a morphism $\varphi: F \to E$ of ...
3
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1answer
1k views

Use Pappus' theorem to find the moment of a region limited by a semi-circunference.

This is part of self-study; I found this question in the book "The Calculus with Analytic Geometry" (Leithold). $R$ is the region limited by the semi-circumference $\sqrt{r^2 - x^2}$ and the ...
3
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1answer
445 views

Maximum cosine for angle between 2 vectors when 1 vector is partially unknown

assuming I have two vectors $A$ and $B$, where $A$ is completely known and from $B$ I know only that the first k components are 0. What is the maximum possible cosine value for the angle between the ...
5
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4answers
706 views

Why do all circles passing through $a$ and $1/\bar{a}$ meet $|z|=1$ are right angles?

In the complex plane, I write the equation for a circle centered at $z$ by $|z-x|=r$, so $(z-x)(\bar{z}-\bar{x})=r^2$. I suppose that both $a$ and $1/\bar{a}$ lie on this circle, so I get the equation ...
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3answers
1k views

How does this equality on vertices in the complex plane imply they are vertices of an equilateral triangle?

I've read that if the complex numbers $a_1$, $a_2$ and $a_3$ are the vertices of a triangle in the complex plane such that $$ a_1^2+a_2^2+a_3^2=a_1a_2+a_2a_3+a_1a_3 $$ then the vertices are actually ...
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0answers
75 views

Translate group definition into geometry system

I need to reword the definition of group (the four axioms: closure, associativity, identity and invertibility) to be lines and points of non-Euclidean geometry (the axiom system defined as geometry). ...
3
votes
2answers
217 views

Find $DF$ in a triangle $DEF$

Consider we have a triangle ABC where there are 3 points, D, E, F such as point D lies on the segment AE, point E lies on BF, point F lies on CD. We also know that center of a circle over ABC is also ...
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4answers
8k views

How to Determine an Equation of a Circle using a Line and Two Points on a Circle

My question goes like this: Determine the equation of a circle tangent to the $x$-axis and passing through $(5,1)$ and $(12,8)$. I need not only the answers, but also the steps on how you did it so ...
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1answer
166 views

Vectors problem

can anyone help me with this problem: Is it possible to construct three vectors (a,b,c) in 3D, such that angle between a and b is 30 degrees, between a and c is 150 degrees, and between b and c is 30 ...
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1answer
115 views

Circle locus, how to satisfy the equation.

$A(-3,1), B(0,-5), P(X,Y)$ If $|AP| = 2|BP|$ prove that $x$ and $y$ satisfy the equation: \begin{aligned} \ x^2+y^2-2x+14y+30 =0 \end{aligned} I get as far as determining the ...
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vote
2answers
1k views

Finding & Plotting equation of hyperbola given foci, and difference in distances between them.

I have to plot the hyperbola (3 of them actually) in MATLAB, and so it'd be good if I could find some sort of general formula. The foci do not necessarily have to be on the axes (e.g. $(5,3)$ and ...
1
vote
2answers
585 views

Stereographic projection of a regular tetrahedron inscribed in the Riemann sphere?

I've been reading about stereographic projections. I did a problem about finding the stereographic projection of a cube inscribed inside the Riemann sphere with edges parallel to the coordinate axes. ...
1
vote
1answer
265 views

How to scale a polyhedron contained a 3-sphere?

In the 3-sphere simulator I am building, the viewpoint is contained in the space of a 3-sphere (the surface of a 4-D hypersphere), and the user is able to navigate through it. There are some ...
3
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1answer
1k views

Equation of a sphere as the determinant of its variables and sampled points

Searching for an equation to find the center of a sphere given 4 points, one finds that taking the determinant of the four (non-coplanar) points together with the variables $x$, $y$, and $z$ arranged ...
3
votes
1answer
178 views

Algebra question about Triangle Interiors

I was reading about Triangle Interiors on Wolfram Alpha: http://mathworld.wolfram.com/TriangleInterior.html and they have a simple equation: $$\mathbf{v} = \mathbf{v}_0 + a\mathbf{v}_1 + ...
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2answers
766 views

Given the cartesian coordinates of four points, how to calculate the interection of two lines they form?

Given four complex numbers $A, B, C, D$ interpreted as points on the plane, how can I calculate the number that represents the intersection of the lines formed by $A, B$ and $C, D$?
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2answers
137 views

is this equation solvable?

Can someone please solve these 2 equations to get values of h and k? I know the values of h and k but not sure how to solved these equations to get h and k 's values $(20.01 - h)^2 + (17.94 - k)^2 = ...
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3answers
114 views

calculating a point on circumference

See the diagram Known values are A: (-87.91, 41.98) B: (-104.67, 39.85) C: (-96.29, 40.92) L: 14.63 // L is OC Known angles ...
2
votes
5answers
2k views

finding center of circle

How can I calculate center of a circle $x,y$? I have 2 points on the circumference of the circle and the angle between them. The 2 points on the circle are $P_1(x_1,y_1)$ and $P_2(x_2,y_2)$. The ...
3
votes
2answers
2k views

Altitudes of a triangle are concurrent (using co-ordinate geometry)

I need to prove that the altitudes of a triangle intersect at a given point using co-ordinate geometry. I am thinking of assuming that point to be $(x,y)$ and then using slope equations to prove ...
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2answers
216 views

Vector Geometry - relation between a point and a line with angle and one known point on it

I have two problems I will be very grateful if somebody helps me about them. If I have a line $L_1$ with a known point $(x_1, y_1)$ on it and has slope $\theta_1$, how do I know if a point $P=(x, y)$ ...
2
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2answers
213 views

Rigorously showing there are infinitely many points of intersection?

I'm working on a problem that states if $k\geq 3$, $x,y\in\mathbb{R}^k$, $|x-y|=d>0$, and $r>0$, then (a) If $2r>d$, there are infinitely many $z\in\mathbb{R}^k$ such that ...
2
votes
2answers
271 views

Conditions for intersection of parabolas?

What are the conditions for the existence of real solutions for the following equations: $$\begin{align} x^2&=a\cdot y+b\\ y^2&=c\cdot x+d\end{align}$$ where $a,b,c,d $ are real numbers. ...
3
votes
2answers
10k views

How do I find the equation of a tangent line to a curve?

I'm given $x^2+2x-4$ at $x=2$ and I have to find the tangent line to this curve at that point...
0
votes
1answer
132 views

lines intersection

I have to find intersection of two lines ($AH$ & $CD$) $A(3.42,-1.84,8.56) $ $B(-3.42,3.84,-8.56) $ $C(0.00,16.25,0.00)$ $AH$ is the perpendicular; $CD$ is the median I tried so: Firstly I ...
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1answer
144 views

Formula to show square root between 2 values

Bear with me - it's been a while since I did this at school! I need to plot a curve in the form of a square root (kind of an 'r' shape if you will) I have 6 intervals along my x axis, and my maximum ...
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1answer
106 views

Mirorring a set of Points

Let's say I have a cloud of points, and I know the equation of the symmetry plane. I'd like to mirror every single point with respect to this plane. It might be much simpler than I think, but I have ...
2
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1answer
379 views

Need help with the proof of conic section

Prove that the intersection of a plane and a object consist of one cone and one upside-down cone where the tip of cone meet is either degenerate conic or conic Also, idenify in what situation, the ...
2
votes
1answer
199 views

What is wrong with this proof that isometries must be surjective?

Let $\phi : \mathbb{R}^2\rightarrow\mathbb{R}^2$ be an isometry. Suppose $\phi$ is not surjective, that is there exists some $v \in \mathbb{R}^2$ whose fiber $\phi^{-1}(v)$ is empty. Then by the ...
0
votes
2answers
141 views

Solving this kind of equation

Say you have two equations with three variables, the first is the equation of the surface of a sphere and the second of a plane. In this case they intersect in a point $(1,0,0)$. The only way I know ...
2
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0answers
476 views

projection of a sphere onto a plane

Consider you have a sphere centered at the origin.The sphere has a diameter of $\frac{1}{2} \sqrt{\frac{3}{2}}$. This means that the inscribed cube has an edge of 1. Take any point from the plane ...
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5answers
1k views

Distance Between A Point And A Line

Any Hint on proving that the distance between the point $(x_{1},y_{1})$ and the line $Ax + By + C = 0$ is , $$\text{Distance} = \frac{\left | Ax_{1} + By_{1} + C\right |}{\sqrt{A^2 + B^2} }$$ What ...
4
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3answers
203 views

Points at integer distance

How many points can one can place in $\mathbb{R}^n$, with the requirement that no $n+1$ points lie in the same $\mathbb{R}^{n-1}$-plane, and the euclidean distance between every two points is an ...
0
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2answers
104 views

Calculate Points for a Parallel Line

Given a line running through p1:(x1,y1) and p2:(x2,y2), I need to calculate two points such that a new parallel line 20 pixels away from the given line runs through the two new points. Edit: The ...
0
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1answer
984 views

Find point on sphere with directional tangent vector

Say a sphere equation like this: $x^2+y^2+z^2=5$. I want to find a point on the sphere whose tangent vector is perpendicular to the vector $\begin{bmatrix} 2\\ 3\\ 4 \end{bmatrix}$. I go ...