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

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Aspect Ratio of Cylinder, Pyramid and Dome

The aspect ratio can easily be defined for rectangular geometries ($AR = height/width$). Is there a definition for aspect ratio of a dome, cylinder, and pyramid (Here standard pyramid and dome were ...
2
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2answers
81 views

Locus of image of point in a line.

I am given the following question: Find the locus of the image of the point $(2,3)$ in the line $$\text{L}:(2x-3y+4)+k(x-2y+3)=0$$ where $k$ is any real number. Attempt at solution. I ...
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1answer
21 views

Coordinate-geometry curiosity question

How can we draw a triangle give one of its vertex and the orthocentre and circumcentre? I tried to invoke the concept of 9 point circle and tried using the centroid but could not succeed in making ...
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1answer
88 views

Find the intersection of two lines passing through given points

Line A goes through the points (4,5) and (-2,-1) and line B goes through the points (3,3) and (6,1). At what point do they intersect? I found the equations of the 2 lines, for A I got: $y = 9-x$, ...
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2answers
56 views

How can one solve this equality geometrically?

Given that $x,y$ are real such that: $$x+2y=\dfrac{1}{2},$$ how can one show, geometrically that $$x^2+y^2\geq \dfrac{1}{20}?$$ I see that $x^2+y^2-\dfrac{1}{20}=5\left(y-\dfrac{1}{5}\right)^2$ ...
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1answer
34 views

Analytic-geometry rotation concept

I am confused how my book comes up with the following formula- Lets consider a Right angled Isoceles triangle with $2$ vertices on hypotenuse given as $(x_1,y_1)$ and $(x_2,y_2)$ Now the 3rd ...
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1answer
79 views

What are the distances from a line to the tangents of a circle?

I have a line given by two points, and a circle given by its origin and radius. I need to find the perpendicular distance between the line and the two tangents of the circle that are parallel to the ...
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1answer
106 views

Find the focus, vertex, latus rectum of the parabola

The problem is, Find the focus, equation of directrix, vertex, length of latus rectum of the parabola given by, $$\left(\alpha x+\beta y+\gamma\right)^2=Ax+By+C$$ I am stuck with the problem for ...
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3answers
467 views

Why this polynomial represents this figure?

I'm trying to understand why this figure is represented by this polynomial expression: My goal is to prove directly why cartesian product of natural numbers is equinumerous to the natural numbers. ...
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2answers
50 views

Normal Vector of plane for Rotation

I reading a code where Normal vector to a plane is given. then a,b,c are taken (what I guess is direction ratio values). ...
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1answer
42 views

Orthocentre of a triangle

How do we determine the orthocentre of a triangle when the vertices are given as $(0,0),(x_1,y_1),(x_2,y_2)$? In a normal case i would take out the equation of any two perpendicular bisectors, get ...
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2answers
107 views

Determining the 3rd vertex of an Equilateral and Right angled isoceles triangle.

I am really having problems solving the following problems: If $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of the two vertices on the hypotenuse of a right angled isosceles triangle then the ...
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3answers
44 views

What will be the other vertex of the triangle?

Two vertices of a triangles are $(5,-1)$ and $(-2,3)$. If the orthocenter of the triangle is the origin, what is the other vertex ? My approach was that since the three vertices and the orthocenter ...
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5answers
2k views

Can area be irrational?

I'm stuck in a question of my book which says: If in an equilateral triangle the coordinates of two vertices are integral then what can we say about the coordinates of the third? The answer is that ...
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2answers
24 views

Graphical transformation

I have a burning question to ask regarding graphical transformation: Suppose I have a function $f(x)$ I want to find $f(ax+b)$ for non zero $a,b$. There are two approaches that I can go: First: ...
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0answers
29 views

Can a line in 3-space have all direction cosines $=\frac{1}{2}$

I immediately found that it is impossible since the squares of the direction cosines have to add to 1 and $3 \times (\frac{1}{2})^2 \neq 1$. However, the textbook asks to "interpret geometrically", ...
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0answers
42 views

Geometrical properties of tetrahedra under rotation

Consider two tetrahedra which share the same point of origin but differ in both scale and rotation over the X-axis. Can someone explain why the following points meet with these parameters? Both have ...
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0answers
21 views

How to plot quadratic forms

I'm studying quadratic forms in algebra at the moment and I've been asked to plot the following curve: $$3x^2+4xy+3y^2-\sqrt{2}x+\sqrt{2}y=1$$ I have used the following transformations: ...
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2answers
24 views

Conics - required to show $SR \times S'R' = b^2$

Consider the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b > 0$. $R$ and $R'$ are the feet of the perpendiculars from the foci $S$ and $S'$ on to the tangent at ...
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0answers
11 views

Pascal and Brianchon's theorems for hyperbolic paraboloid

How would one formulate a version of these two theorems for the hyperbolic paraboloid, and what would be a simple proof? How are the classical formulations of these theorems related to this quadric?
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0answers
10 views

Prove quadrics are rational algebraic surfaces.

I have to prove that an irreducible quadric in RP^3 is a rational algebraic surface, ie, the homogeneous coordinates of any point can be expressed as polynomials in two variables. My idea was to do ...
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0answers
15 views

Quadratic singularities and local curves.

Everything is to be understood over the complex field. Assume you have two finite dimensional $\mathbb{C}$-vectorial space $V$ and $W$. You are given a bilinear form : $$\phi:V\times V\rightarrow W ...
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1answer
61 views

Finding quadric's rotation matrix

I want to know how to find the rotation matrix of a quadric in general and the eigenvalues and eigen-vectors, in particular I am given $$ Q(x,y,z)=18x^2+9y^2+14z^2+8xy+8xz−4yz−2x−6y−14z+6=0 $$ and I'm ...
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1answer
12 views

General versions for quadrics of Pascal and Brianchon theorems

I am looking for a generalization to quadrics (with proofs) of Pascal's and Brianchon's theorems. It´s for Three dimensional analytical geometry. I would be very thankful if you could point me ...
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16 views

Quadric generalizations of Pascal and Brianchon's theorems

I am looking for a generalization to quadrics (with proofs) of Pascal's and Brianchon's theorems. It´s for Three dimensional analytical geometry. I would be very thankful if you could point me ...
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1answer
39 views

Find the rotation matrix of a quadric

I want to know how to find the rotation matrix of a quadric in general and the eigenvalues and eigen-vectors, in particular I am given $Q(x,y,z)= 18x^2+9y^2+14z^2+8xy+8xz-4yz-2x-6y-14z+6=0$ and I'm ...
2
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0answers
62 views

Analytic-geometrical properties of dodecahedron

Consider the following projection of a dodecahedron: An equilateral triangle can be projected to make points $A, B, C, D, E, F$ intersect with it's edges. What would be the mathematical proof (if ...
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1answer
51 views

Algebraic Geometry Approach To Study The Surfaces Given The Intersection Curve

I'm NOT a Mathematician and I'm totally new to the field of Algebraic Geometry. A friend of mine told me that one thing which is studied in this field is to consider a curve as a set of points in n-D ...
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3answers
110 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 ...
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2answers
36 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 ...
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0answers
213 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 ...
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3answers
111 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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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 ...
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2answers
75 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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1answer
50 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} ...
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1answer
31 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 ...
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1answer
52 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 ...
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1answer
36 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
58 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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3answers
140 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 ...
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4answers
264 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. ...
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1answer
91 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, ...
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3answers
667 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 ...
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2answers
69 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 ...
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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 ...
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1answer
234 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: $$\frac{ax}{x^2+y^2}+\frac{by}{x^2+y^2}+c=0$$ ...
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1answer
40 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
34 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: ...
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1answer
26 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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62 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 ...