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

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5
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3answers
97 views

There is a unique quadric through three disjoint lines

There is a classical exercise that three disjoint lines in $\mathbb{P}^3$ are contained in a quadric surface $Q$. The existence is trivial. Every quadric in $\mathbb{P}^3$ is determined by nine ...
0
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2answers
29 views

Regular polygons inscribed in an ellipse

A regular $n$-gon is inscribed in an ellipse which is not a circle. what are the possible values for $n$? I know I can inscribe a square or even a equilateral triangle, but can we do it for all $n$? I ...
9
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0answers
179 views

Mathematical properties of two dimensional projection of three dimensional rotated object

Please be gentle as I do not have any degree in maths. By using a compass/straighedge method to construct Metatron's cube, a regular dodecahedron can be inferred from intersecting points. I'm looking ...
3
votes
3answers
95 views

Tangent surface of a twisted cubic curve

I am trying to describe the tangent surface to a twisted cubic curve $C$, i.e. the curve which is given parametrically by $t\mapsto(t, t^2, t^3)$. This surface $S$ is given parametrically by ...
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vote
2answers
37 views

finding $AX\cdot BX\cdot CX\cdot DX\cdot EX\cdot FX\cdot GX\cdot HX$

Given a Regular Octagon $ABCDEFGH$, $AE=2$. On $AE$ we choose point $X$ which dividing $AE$ in the ratio of $3:1$. Need to find: $AX\cdot BX\cdot CX\cdot DX\cdot EX\cdot FX\cdot GX\cdot HX$ Any ...
4
votes
2answers
68 views

Finding axis of ellipse described by $x=a\cos t+ h\sin t$,$ y=b\sin t + g\cos t$

Hi I am in need of help here for my project. Basically I have managed to obtain this form of equation. Example: $a=-181,h=33,b=185.9$ and $g=18.3$. When I plot it on a graphing program, it looks like ...
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votes
1answer
37 views

Intersection between a line and a plane.

A line can either lie on a plane, lie parallel to it or intersect it. Determine, if there is one, the point of intersection between the line given by the equation $$\displaystyle\frac{x−5}{2} ...
0
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1answer
25 views

Help Deriving the Midpoint Formula

One of the problems in one of the packets that I'm going through to review for a pre-test for an independent-study calculus class has asked me to derive the midpoint formula. I've gotten to the point ...
5
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0answers
42 views

Upper bound on the distance of orthogonal matrices

Dear math stackexchange users, I have a question on orthogonal matrices: suppose I have a matrix $X\in\mathbb{R}^{n\times n}$ and I consider the orbit of the orthogonal group $O(n)$ acting from the ...
2
votes
1answer
33 views

Relation between a differential equation satisfied by parabolas and a formula for the slope of their tangents

Statement 1: The slope of the tangent at any point P on a parabola, whose axis is the axis of x and vertex is at the origin, is inversely proportional to the ordinate of the point P. Statement 2: The ...
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0answers
38 views

Equation of common tangent(s) to two parabolas

Suppose we are given two parabolas, described by their directices $d_1: a_1x+b_1y+c_1=0, d_2: a_2x+b_2y+c=0$ and foci $F_1(p_1,q_1), F_2(p_2,q_2)$. How does one find equations of common tangents to ...
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votes
3answers
118 views

How can the centers of these 5 related circles be specified as a formula?

This is my first time posting in this forum, so please forgive me if my question is too involved or if I've posted it in the wrong area. I hope someone finds it interesting enough to try their hand at ...
10
votes
4answers
230 views

Tricky 3d geometry problem

We have a cube with edge length $L$, now rotate it around its major diagonal (a complete turn, that is to say, the angle is 360 degrees), which object are we gonna get? Astoundingly the answer is D. ...
4
votes
1answer
46 views

How to get coordinates of a point after an image is rotated? (with images)

I have a problem that involves a rotating image and finding a previously known point. Firstly, there is a sequence with the rotation. We start with an empty image. A line is drawn vertically, ...
2
votes
4answers
300 views

Find the coordinate of third point of equilateral triangle.

I have two points A and B whose coordinates are $(3,4)$ and $(-2,3)$ The third point is C. We need to calculate its coordinates. I think there will be two possible answers, as the point C could be on ...
0
votes
2answers
46 views

Finding the coordinates of a parallel line given line coordinates and a distance

I have a path defined by a list of (x, y) coordinates and I want to create two additional paths, one offset by a distance of 0.25, the other by -0.25. I think that could be done by finding parallel ...
0
votes
1answer
17 views

Find the equation of the hyperbola that satisfies this condition

Focus is at $F\equiv(−3−3√13, 1)$, asymptotes intersect at the point with coordinates $(−3, 1)$ and one asymptote passes through $(1, 7)$ I've solved some problems that involve equations of ...
3
votes
1answer
221 views

Simplifying an equation (circle)

I'm trying to work on a problem, and I'm stuck at simplifying this equation. I do not why I cannot see it: so the book gives the following equation: $$\frac{ax}{x^2+y^2}+\frac{by}{x^2+y^2}+c=0$$ ...
0
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1answer
39 views

Why 9 points determine a quadric

The books I have state this redult as obvious from the definitions, but it is not clear to me why this is so.
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0answers
28 views

Find the direction from which the projected area of a loop is maximal

How do I find the direction from which the projected area of a loop is maximum? Should I try to use intuition or is there a simple mathematical way to find it? The problem given was the following: ...
0
votes
1answer
16 views

How do I find the symmetrical point B given the centre of symmetry C and another point A?

I have a point $A (-2k; 3)$ and a point $B$ that is symmetrical to the point A given the centre of symmetry $C (-1; 0)$. I tried applying the following formula, where $x_o$ and $y_0$ are the ...
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0answers
59 views

A problem about the intersection of a plane and a sphere

So I have been trying to solve this (rather basic) geometry problem but don't know how to parametrize the functions and get an answer (please check below). Ok, so my problem reads like this: prove ...
0
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1answer
46 views

Given one endpoint and midpoint in (x,y) of a line segment, explain how to find the other end point.

A line segment with one end at C(6,5)has midpoint M(4,2). Determine the coordinates of the other endpoint, D. Explain your solution and describe a method to check your answer.
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votes
6answers
207 views

Area enclosed by the graph of $13x^2-20xy+52y^2+52y-10x=563$.

Find the area enclosed by the graph of $13x^2-20xy+52y^2+52y-10x=563$. First I saw that this cannot be a circle ($xy$ term), and it cannot be an ellipse with axes parallel to the coordinate axes. But ...
5
votes
6answers
93 views

Why is the equations for a perpendicular line $-\frac{1}{m}$?

Why is it just $-m$? Lets say $y=x$ and the $y$ intercept is at $0$ if we created another line that was $y=-x$, wouldn't that make it perpendicular? Note: Don't exactly know what tags to use, feel ...
3
votes
1answer
59 views

How homogenization of line and curve works?

I am given a curve $$C_1:2x^2 +3y^2 =5$$ and a line $$L_1: 3x-4y=5$$ and I needed to find curve joining the origin and the points of intersection of $C_1$ and $L_1$ so I was told to "homogenize" ...
5
votes
2answers
102 views

Rigorous books on geometry

I am looking for a rigorous book on both 2d and 3d euclidean geometry, and also how analytic geometry can be developed from synthetic geometry. I haven't really found such a book yet. I would be very ...
0
votes
1answer
18 views

How do I determine k so that the line of the beam is parallel to a $60^\circ$ angle?

I have the equation of a beam that looks like this: $$(x + y - 5) + k(2x - 3y) = 0$$ I know that the angular coefficient of a $60^\circ$ angle is equivalent to the root of 3. $$m = \sqrt3$$ Though, ...
0
votes
1answer
65 views

formula for a sphere?

is there such a thing as a formula for a sphere? Is it $x^2+y^2+z^2=1$? if so, does the $1$ denotes a radius of $1$ for said sphere? what are the possible alterations for such a formula?
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0answers
38 views

Proving That Two Paths of Different Lengths Are Adjoined

In the section on 'Adjoining Paths' of its 'Topology' book's page on 'Path Connectedness,' WikiBooks shows that, for any topological space $X$ with members $a$, $b$, and $c$, the following…: ...
0
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1answer
18 views

Derivation of the Parametric Form of the Equation of a Line From Its Two-Point Form?

Wikipedia's documentation on the parametric form of a linear equation states in the paragraph between two different sets of equations available for use in determining a line's parametric equations ...
0
votes
3answers
47 views

What is the equation of the line that is parallel to the y-axis?

I have a line, parallel to the $y$-axis, that passes through a point, P: $$P(1/2,-3/5)$$ What is the equation of the line? What I tried: $$(y−y_0)=m(x−x_0)$$ $$(y+3/5)=m(x−1/2)$$ ...
0
votes
1answer
40 views

How small can an external angle of a circumference be if made of tangents?

Lets imagine the angle ABC where the lines AB and CB are tangents to a circumference which center is C. Lets assume that the points where the line AB touches the circumference is P and the point where ...
3
votes
2answers
80 views

Show that $PF.PG=b^2$ in a hyperbola

If the normal at P to the hyperbola $\frac {x^2}{a^2}-\frac {y^2}{b^2}=1$ meets the transverse axis in G and the conjugate axis in G' and CF be the perpendicular to the normal from the center C then ...
0
votes
1answer
34 views

What are the coordinates of your position?

Suppose you start at the origin, move along the x-axis 3 units. Then face downwards and move forward 4 units. Then turn right and move 7 units. Then (relative to your current position) face downwards ...
3
votes
3answers
118 views

Property of ellipses involving normals at the endpoints of a focal chord and the midpoint of that chord

While solving a book on ellipses, I came across the following property of an ellipse which was given without proof :- If the normals be drawn at the extremities of a focal chord of an ellipse, a ...
0
votes
2answers
44 views

Finding the value of $p$ in the parabola $y^2=2px$

I just started to learn the parabola shape and I have a question: Given the parabola $y^2=2px$ $(p>0)$. The chord $AB$ of the parabola passes through the focus $F(\frac{p}{2},0)$. The slope $m$ ...
0
votes
1answer
18 views

Finding the distance of the line to apoint

Find the distance from $3x-4y-10=0$ to the point $(2,0)$ my answer here is $ \dfrac{-4}{2}$ or $-2$ by substituting the given by the use of the formula but Im just wondering if there's a negative ...
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votes
1answer
63 views

Find equations of two circles drawn through the origin which cut another circle orthogonally and touch a line

Find equations of two circles which are drawn through the origin to cut the circle $$x^2+y^2-x+3y-1=0$$ orthogonally and to touch the line $$x+2y+1=0$$. $$x^2+y^2-2ax-2by=0$$----(1) is the general ...
0
votes
1answer
40 views

Meaning of $\dashv\vdash$

I was looking at ProofWiki's articles 'Definition:Equidistance' and 'Definition:Between (Geometry)'and came across the symbol '$\dashv\vdash$.' What does it mean?
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vote
1answer
34 views

Do we have $proj_u(a) + proj_u(b) = proj_u(a+b)$?

Let $a, b, u$ be vectors in $\mathbb{R}^3$. For two vectors $r, u$ in $\mathbb{R}^3$, let $proj_u(r)$ be the projection of $r$ on the line of $u$ in $\mathbb{R}^3$. Do we have $proj_u(a) + proj_u(b) = ...
3
votes
1answer
107 views

Intersection of a line through two points on a unit circle with real axis

Suppose we are given two points on unit circle which are represented as complex numbers $u$, $v$. We want to show that the intersection of the line through $u$ and $v$ and the real axis is ...
0
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0answers
51 views

Good book for Solid Analytical Geometry?

So my teacher uses this book, William H McCrea's Analytical Geometry of Three Dimensions, but it's awfully hard and dry. I need something with more exercises and better explanations, but that covers ...
0
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0answers
34 views

Is the unit circle uniquely defined by all it's sliced averages?

Suppose you have a unit circle $g(x,y) = x^2+y^2-1 = 0$ and for each $\theta \in (-\pi/2, \pi/2)$ you associate a map $f_\theta(c): R \to R^2$ defined by $$f_\theta(c) = \langle g^{-1}(0) \cap ...
3
votes
2answers
102 views

Equation for Tangent Line that passes through $(0,1)$ on the curve $y = \ln x$

I'm totally lost. I've been trying to figure this out. This is what I've figured out: $dy/dx = 1/x$ $y$-intercept $= 1$ So I try to do $y-y_1 = m(x-x_1)+b,$ which I get as $y-1 = 1/x(x-0)+1,$ ...
0
votes
2answers
60 views

Finding the length from an interior point of a triangle to a vertex given distances to the other two

So let's assume that there is a triangle ABC and there is a point P inside of ABC. You are given the distances of AP and BP and you are trying to solve for CP. I faintly remember reading something ...
4
votes
4answers
97 views

Find the equation of a circle, given a point on it and a point where it is tangent to a given line

The given question is: Find the equation of the circle that passes through point $(-3,-4)$ and touches the line $x-y+7=0$ at the point $(-5,2).$ What I did was: Took the given points $(-5,2)$ and ...
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votes
1answer
18 views

What is the domain of the given function with the greatest integer?

The domain of the function $$f(x)=\sqrt{\frac{4-x^2}{[x]+2}}$$ where $[x]$ represents the greatest integer function, is (a) $(-\infty,-1)\cup[-1,2]$ (b) ...
0
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0answers
34 views

Arc measures in a circle

Suppose we have a quadrilateral inscribed in a circle prove that angles inside the same arc are equal
6
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1answer
103 views

A parabola lemma

I am looking for a previous reference and/or a geometric proof of the following lemma: Let $P$ be the parabola $y=x^2$. Let $a$, $b$, $c$, $d$ be four points on $P$ sorted from left to right, and let ...