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

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To prove by vector method , non-parallel sides of a trapezium having equal diagonals are equal.

How do we prove by vector method that "if the diagonals of a trapezium have equal length then the non-parallel sides of the trapezium have equal length." ? (taking $ABCD$ to be the trapezium with ...
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51 views

Determine properties of a shape defined by a function?

I have a number of functions that define 3d shapes. They take a point $(x, y, z)$ as a parameter and return a real value representing information about the shape. On the surface of the shape: $f(x) ...
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2answers
155 views

Is there a real continuous function of a Cube?

Sorry if this seems like a basic question, but I'm having troubles finding an answer in my research. I'm looking for a function that if given a test point (x, y, z) will return a real value describing ...
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0answers
55 views

points possible on a circle

Let $A, B, C, D$ and $E$ be five points marked in clockwise order, on the unit circle in the plane (with centre at origin). Let $\alpha$ and $\beta$ be real numbers and set $f(p)=\alpha x+\beta y$ ...
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1answer
110 views

How do I make pi = 3?

This question emerges from a discussion on quora which concluded that if a circle was drawn on the surface of a sphere, the ratio of radius (from the circle's centre as projected to the sphere's ...
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2answers
348 views

Books on geometric transformations and/or analytic geometry?

I've been looking to expand my knowledge in geometry as it's not covered in my undergraduate curriculum. For some reason I'm repelled by the classical approach (hopefully it will pass) as I feel it's ...
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4answers
359 views

Direct method to find the equation of a circle.

Suppose we are given four concyclic points or two lines which intersect the axes in concyclic points. Many a times, one point has a variable as a co-ordinate. Suppose the concyclic points are ...
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1answer
53 views

Rays in fat parabolas

Let $\epsilon>0$. Let $F\subseteq \mathbb{R}^2$ be the set of all points that lie at a distance less than $\epsilon$ from the curve $y=x^2$. Can $F$ contain a ray? That is, is there (for some ...
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103 views

Given Two Lines, How to Find a Point on Line $2$ given a Specific Distance From Line $1$?

I have two lines (U and V). What is the method to calculate a point on V given a specified distance (d) from U? The lines may be assumed that they do intersect (are not parallel) and are straight ...
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3answers
135 views

Proof that a line cuts in half the area of a parallelogram iff it goes through the intersection of the diagonals?

I read a theorem in a book which says that a line bisects a parallelogram iff it goes through the intersection of the diagonals. The edge case of this is of course if the line is one of the diagonal ...
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2answers
55 views

Proving Coplanarity of 3 vectors

Let $a,b,c$ be three vectors such that $|a|=|b|=|c|=\sqrt{2} $ and $a\cdot b = b\cdot c = a\cdot c = -1 $ . How can I prove that they are all coplanar? I found that the angle between every two of ...
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156 views

Coordinate Transformations

I am physics student. My mathematical background is quite weak. I just want to know the similarities (if there are any) between coordinate transformation of two kinds : Rotation of coordinate (and ...
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3answers
88 views

prove that the rose (in the polar plane) has $2n$ “petals” when $n$ is even

prove that the rose $r=\cos(n\theta)$ (in the polar plane) has $2n$ "petals" when $n$ is even. How can I start this demonstration? I would appreciate your help
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145 views

How to calculate center coordinates of two reverse arcs in 3D space

Given 3D points P1(200,60,140), P2(300,120,110), P3(3,0,-1), P4(-100,0,-1) and the radius of arc C1MP3 is equal to radius of arc C2MP1. How do I calculate coordinates x, y, z of points C1 and C2? ...
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1answer
249 views

The formula for pitch circle diameter.

I want to put $n$ number of circle with $r$ radius each in a big circle. Want to calculate the radius $R$ of the big circle. How can this be achieved?
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1answer
60 views

Finding the locus of the following conditions

Find the locus of all points in the plane, whose distance from a constant point $F=(x_0,y_0)$ divided by their distance from the vertical line $L=\{(k,y)\mid y \in \mathbb R \}$ equals a constant ...
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1answer
58 views

Finding affine transformation

Find affine transformation which takes the ellipse $x^2+4y^2+2x-8y+3=0$ to the form of the ellipse ${x^2 \over 9}+{y^2 \over 16}=1$. So I took the quadric and reached to a standard form: ${(x+1)^2 ...
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1answer
122 views

translate coordinates on circle to percentage?

I'm coming more from a programming point of view but the question is pure math. The only strange thing, I guess, is that the coordinate system is like this: ...
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1answer
672 views

Equation of circle with given radius passing through two given points

Find the equation of the circles passing through two points on the $y$ axis at distances 3 units from the origin and having radius 5. (This a homework problem but I do not know how to solve it.)
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1answer
88 views

Find the equation of the hyperbola?

The hyperbola being an orthogonal parabola, for which $(-1,2)$ is a focal point and $x-y+1=0$ is an asymptote. If I have the equation for the asymptote $y=x+1$ is the center $(0,1)$? I do not know ...
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1answer
45 views

Find equation of hyperbola?

The hyperbola has center $(0,0)$, and goes through the points $(3,1)$ and $(9,5)$ and the coordinate axes are the symmetry axes. The correct answer is $x^2 - 3y^2=6$.
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1answer
92 views

General Coordinate Geometry Problem - How to deal with lines parallel to y - axis

In coordinate geometry, whenever we solve a problem we see that if the resulting solution is a line, then all the lines which are parallel to y - axis are left out since their slope will be $\infty$ ...
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2answers
51 views

An equation for an ellipse

Definition: An ellipse is a curve on a plane surrounding two focal points such that a straight line drawn from one of the focal points to any point on the curve and then back to the other focal point ...
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1answer
84 views

Unicursal Curve Double Points

To quote Goursat: It is shown in treatises on Analytic Geometry that every unicursal curve of degree n has $\frac{(n-1)(n-2)}{2}$ double points, and, conversely, that every curve of degree n ...
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1answer
113 views

Analytical Geometry problem with complex numbers - alternate solutions.

The question is to show that the equation of the lines making angles $45^\circ$ with the line: $$ \bar{a}z + a\bar{z} + b = 0; \;\;\;\;\; a,z \in \mathbb{C}, b \in \mathbb{R} $$ and passing through a ...
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391 views

Can we plot a regular octagon on a set of axes, where all vertices of the octagon lie on integer co-ordinates?

I'm a high school teacher and someone asked me this in my class, and to be honest I'm quite stumped! I haven't done any high level math in such a long time, and I'm really not sure how to approach ...
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2answers
76 views

Recognize conics from the standard equation

Suppose $Ax^2+Bxy+Cy^2+Dx+Ey+k=0$ is a conic in the Euclidean plane. How do I recognize what is it? In my book they have proved the determinant test that if $B^2-4AC$ is $>0$ if hyperbola, $=0$ if ...
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3answers
112 views

2 dimensional coordinate geometry

If $L_1$ and $L_2$ are two lines belonging to the family of lines $(3+2s)x+(4+3s)y=7+5s$ such that they are at maximum and minimum distances from the center of the circle $3x^2 +3y^2 -12x-18y-91=0$, ...
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1answer
160 views

Show that the focus of the parabola lies on the nine point circle of the triangle.(dificult)

A parabola is drawn such that each vertex of a given triangle is the pole of the opposite side;show that the focus of the parabola lies on the nine point circle of the triangle and that the ...
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0answers
80 views

Clarification of the Jacobian

Well, that was cool if not tedious but I understand the Jacobian and its application to changing coordinate systems. $${J_{POLAR}= \rho}$$ $$ {J_{cyl}= \rho}$$ and $${J_{sphere}=\rho^2\sin\phi}$$ ...
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1answer
133 views

Why is the Jacobian ${\rho}$

Just a little confused. When I find the volume of a cone (or a sphere) for that matter I multiply the partial derivatives by the Jacobian. ${\rho}$ for a cone. and ${\rho^2 \sin \phi}$ for a sphere. ...
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2answers
51 views

Proving the following is a Group

I'm studying this weird course called "Analytic Geometry", but in reality it seems like a mash of modern or abstract Algebra (...I'm not so sure...), and includes stuff like Affine transformations, ...
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2answers
2k views

Generalizing the hardest question on the practice math GRE

The most-missed question on the Math GRE is the following: How many times does $x^{12}$ intersect $e^x$? Because I told you it was hard, you probably realized it was a trick and got the right ...
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1answer
112 views

Trying to understand Volume of a cone without the unit sphere

I have been working on the double integral proof for the volume of a cone. I found that I can use a unit-sphere Where the base of the cone is the equator and the height is the distance ${\rho}$ to ...
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1answer
2k views

Find Equation of a Perpendicular Line Going Through a Point

I have the following parametric equation for line g: $$ x=3t\land y=-7+5t\land z=2+2t $$ I have to find the equation of a line perpendicular to $g$ and going through point $Q(3,-2,4)$ which lies on ...
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65 views

A point can be viewed as a circle?

In analytic geometry, can a point be viewed as a circle? In analytic geometry, can the point $(0, 0)$ be view as the circle of zero radius with center $(0, 0)$?
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Calculating the coordinates of a vertex within a triangle so that the triangle becomes a right-angle

I have three points defining the vertices of a triangle A(1,4) B(6,7) C(5,1) I have found that vector AC has a slope(m) of 0.6 and vector BC has a slope of 6. From these slopes have the angles of ...
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1answer
564 views

Line equation and distance between a line and a point in 3D space

I've looked for many similar questions but could not find an answer that helps. I have a point $p$ $(x,y,z)$ on a plane, and a normal to the plane $n$ $(a,b,c)$. I need to find the equation of a line ...
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1answer
140 views

Ellipse, hyperbola and principle axis

Would anyone mind telling me how to solve (a)? I have no idea what I should do to solve this problem. Also, what is principal axes?
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1answer
1k views

Finding a polar equation for an ellipse

I need to find the polar equation of an ellipse, with one of its foci at the pole (origin), with a horizontal major axis of $10$ units and a vertical minor axis of $6$ units. So first off, the ...
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1answer
49 views

When are $ r_1: ax + by + c=0 $ and $ r_2: a'x + b'y + c'=0$ distinct?

When are $ r_1: ax + by + c=0 $ and $ r_2: a'x + b'y + c'=0$ distinct? I think, if $$rank\left(\begin{array}{cc} a & b \\ a' & b' \end{array} \right)\neq 1$$ then $r_1$ and $r_2$ are ...
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1answer
261 views

Sum of euclidean norms with box constraints

minimizing the sum of euclidean norms with box constraints I am a graduate student in computer science, making a thesis on uncertainty geometry. During my thesis I came across the following ...
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0answers
35 views

Finding the point of concurrence of a family of straight lines

If $6a^2-3b^2-c^2+7ab-ac+4bc=0$, then the family of lines $ax+by+c=0$ is concurrent at $(-2,-3)$ $(3,-1)$ $(2,3)$ $(-3,1)$ Multiple answers are possible. I am not able to group terms of the ...
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44 views

Linked comprehension on straight lines

The vertex $A$ of triangle $ABC$ is $(3,-1)$. The equations of the median $BE$ and the angular bisector $CF$ are $x-4y+10=0$ and $6x+10y-59=0$ respectively. Then 1:$\;\;\;$The equation of $AB$ must ...
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1answer
33 views

What type of Math and how to solve (points and distance)

I think it has to do with geometry and algebra. We have a rectangle with four points. We know each point as $(0,0), (0,10),(10,10)$ and $(10,0)$. In the rectangle we have another point $(x,y)$. ...
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1answer
34 views

Find $\vec{QB}$ in terms of $\mathbf c$

I've managed to work out “$\vec{AM}$ in terms of $\mathbf a$ and $\mathbf b$” to be $3\mathbf a+\mathbf b$. But how can I work out “$\vec{QB}$ in terms of $\mathbf c$”?
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59 views

Suppose that $n(r)$ denotes the number of integer points on a circle of radius$>1$…

Suppose that $n(r)$ denotes the number of points with integer co-ordinates on a circle of radius $r>1$. Prove that, $n(r)<2\pi r^{2/3}$ I could not get much help from a similar question, ...
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1answer
398 views

Question about a pair of straight lines

Find the centroid of the triangle formed by the pair of straight lines $12x^2-20xy+7y^2=0$ and the line $2x-3y+4=0$. My doubt is: The given pair of straight lines and the third line all pass through ...
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75 views

Stereographic projection of the icosahedron and snub cube?

Using a steoreographic projection, the three equations associated with the icosahedron with unit circumradius, inradius, and midradius (respectively) are, $$f=z^{20} - 228z^{15} + 494z^{10} + 228z^5 ...
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147 views

Can't find the derivation ${\rho^2\sin\phi}$

I have accepted that the equation of a sphere in spherical coordinates is ${\rho^2\sin\phi}$. The triple integral is just to nice. What I don't understand is what happened to ${\theta}$. How can you ...