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

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Text explanation: Ellipses and their intersection points

Given two ellipses $E_1,E_2$ of equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=4$$ prove that for all $p\in E_1$ there exist a unique ellipse $F_p$ that meets $...
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2answers
26 views

proving that triangles $ABC$, $A'B'C'$ are congruent

Given $AD$ is a median to $BC$ in triangle $ABC$, and $A'D'$ is a median to $B'C'$ in triangle $A'B'C'$, and $AD=A'D', AC=A'C', AB=A'B'$. How can i prove that triangles $ABC$, $A'B'C'$ are congruent? ...
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Alternate form for the vector field $x^3\cdot\hat{x} + y^3\cdot\hat{y}$

Is there an alternate form for the vector field $x^3\cdot\hat{x} + y^3\cdot\hat{y}$ in which we can write all in function only of the radius $r=\sqrt{x^2 + y^2}$ ? Thank you
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1answer
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If tangents are drawn from two points which are equidistant from given point, then find the locus

Tangents are drawn to the circle $x^2+y^2=a^2$ from two points on the $X$ axis equidistant from the point $(k,0)$ prove locus of their intersection is $ky​^2=a^2(k-x)$. If I take points as $(k+\alpha,...
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2answers
33 views

Given $\Vert \vec{u} \Vert$ and $\Vert \vec{v} \Vert$ and $\angle 120^\circ$ find volume with sides $\vec{u} \times \vec{v}$, $\vec{u}$ and $\vec{v}$

I am given the following problem: Knowing that $\Vert \vec{u} \Vert = 3$ and $\Vert \vec{v} \Vert = 4$ and also $\angle (\vec{u}, \vec{v}) = 120^\circ$ find the volume of the parallelepiped with ...
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2answers
70 views

If $x^2=\lambda$, then find the value of $\lambda$

A Circle $C_1$ is drawn having any point $P$ on $X$- axis as its centre and passing through the centre of the circle $C: x^2+y^2=1$. A common tangent to $C_1$, and $C$ intersects the circle at ...
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1answer
34 views

Finding straight lines

I am new to analytical geometry and excuse me for my notations. We have four lines: $l_1: u_1x + v_1y + r_1 = 0$ $l_2: u_2x + v_2y + r_2 = 0$ $m_1: u_3x + v_3y + r_3 = 0$ $m_2: u_4x + v_4y + r_4 = ...
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1answer
35 views

How to prove the parallel projection of an ellipsoid is an ellipse?

Take the following ellipsoid in implicit form as an example: $$x^2 + 2 y^2 + 3 z^2 + x y + y z - 2 xz = 5$$ which shows: The parallel projection of the ellipsoid onto $xoy$ coordinate plane can ...
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Great mathematical fusions in math history

Development of the mathematics resembles usually a growing tree - from old branches grow new ones. However sometimes domains of mathematics which were separated for the long time are fused together ...
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1answer
30 views

Find the centre of a circle circumscribing the triangle whose angular points are $(1,1), (2,3), (-2,2)$

The main question is as follows : Find the point $P$, such that $P$ is the centre of a circle circumscribing the triangle whose angular points are $(1,1), (2,3), (-2,2)$. My method : I am ...
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2answers
560 views

How to solve such an equation ? (Line-Plane Intersection)

I don't know how to solve such an equation: $$ t = - \frac{ \begin{vmatrix}1&1&1&1\\x_0&x_1&x_2&x_4\\y_0&y_1&y_2&y_4\\z_0&z_1&z_2&z_4\end{vmatrix}}{\...
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+50

'Tetrahedral' coordinates in space (generalization of hexagonal coordinates)

The Cartesian coordinates are the most widely used in Euclidean space of any dimension. However, there is another set of coordinate systems which can in some way be considered optimal. Imagine ...
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1answer
91 views

Is the smallest ellipsoid enclosing a convex set unique?

Let $S \subset \mathbb{R}^n$ be a convex set. Assume that it is bounded. We want to find an ellipsoid $E$ of smallest volume such that $S \subset E$. Is $E$ unique?
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1answer
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Standard form of an Ellipse Given…

So I'm currently stuck on how to get the standard form of the equation of the ellipse given the characteristics Vertical Major Axis and passes through the points ( 0,6 ) and ( 3,0 ) Any Ideas? ...
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1answer
36 views

Radius of the sphere inscribed at the corner of an irregular polyhedron

Does anyone know the mathematical formula to calculate the radius of a sphere that can be inscribed at the corner of an irregular polyhedron ? There can be several radius, but the sphere should not ...
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1answer
1k views

IMO 2016 Problem 3

Let $P = A_1 A_2 \cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $...
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1answer
396 views

Intersecting two parabolas and computing the angle between the tangents in a point of intersection

I was solving some problems on parabola. I saw a question and solved it, but my solution was way too big. The question was: If $$\left(\frac{a}{b}\right)^{1/3}+\left(\frac{b}{a}\right)^{1/3} = \frac{\...
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1answer
398 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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1answer
48 views

Circle Problem:Which of the following are true

If the circle $x^2+y^2+2gx+2fy+c=0$ cuts the three circles $x^2+y^2−5=0$, $x^2+y^2−8x−6y+10=0$ and $x^2+y^2−4x+2y−2=0$ at the extremities of their diameters, then which of the following are true ? $c=...
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0answers
11 views

Generating an Equation for Quadratic Bezier Curve

So I know that a standard equation for a quadratic Bezier Curve is: $C(t)=(1-t)P_0+2t(1-t)P_1+t^2P_2$ I have been asked to generate an equation to model my quadratic bezier curve. I known that my $...
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1answer
25 views

Find the coordinates of P and Q.

A line is drawn through the point $A(1,2)$ to cut the line $2y=3x-5$ in $P$ and the line $x+y=12$ in $Q$. If $AQ=2AP$, find the coordinates of $P$ and $Q$. From: Mathematics, The Core Course for A-...
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1answer
26 views

Is there a simple way to decide if a hyperboloid is one-sheeted or two-sheeted, given the quadric equation?

Let us say that we have a quadric equation, whose solution set lies in $\mathbb{R}^3$, and you know it's a hyperboloid. Is there a way to analytically decide through a criterion if the hyperboloid is ...
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6answers
80 views

Checking nature of angles of a triangle given the equations of the three lines that form a triangle

Suppose we have three lines $\ell_i=a_ix+b_iy=c_i$, $i=1,2,3$ and we are given that they form a triangle. I need to find which angles are acute and which are obtuse without plotting the lines ...
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22 views

Clarification Needed:Equation Of Refracted Ray/Line

A ray of light is sent along the line $2x-3y=5$.After refracting across the line $x+y=1$ it enters the opposite side after turning by $15^0$ away from the line $x+y=1$.Find the equation of the ...
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2answers
9 views

Given $\angle (\vec{u} , \vec{v}) = 30^\circ$, $\Vert \vec{w} \Vert = 4$ and $\vec{w}$ is $\perp$ to both find $\vec{u} \cdot \vec{v} \times \vec{w}$

I am given the following problem: Knowing that the angle between the unit vectors $\angle (\vec{u} , \vec{v}) = 30^\circ$, $\Vert \vec{w} \Vert = 4$ and that $\vec{w}$ is orthogonal to both of ...
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1answer
14 views

Given the tethraedron $OABC$ find ratio between its Volume and $\vert \vec{OM} \cdot \vec{ON} \times \vec{OP} \vert$

Given the tethraedron $OABC$ where $\vec{OA} = \vec{a}$, $\vec{OB} = \vec{b}$ and $\vec{OC} = \vec{c}$ and the points $M$, $N$ and $P$, which are the midpoints of segments $\vec{AC}$, $\vec{AB}$ and $\...
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2answers
30 views

Sum of slopes between points of concurrent normals to hyperbola is zero

Let $ A_1, A_2, A_3, A_4 $ be four points on the hyperbola $xy = 1$. Suppose that the normals to the hyperbola at these four points are concurrent, i.e. they intersect in a single point. Prove that ...
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1answer
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Given that $ \{ \vec{u}, \vec{v} \}$ are l.i. prove that if $ \vec{w} \times \vec{u} = \vec{w} \times \vec{v} = \vec{0}$ then $\vec{w} = \vec{0}$

I am asked to elaborate on the following proof: Given that $ \{ \vec{u}, \vec{v} \}$ are linearly independent prove that if $ \vec{w} \times \vec{u} = \vec{w} \times \vec{v} = \vec{0}$ then $\vec{...
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2answers
50 views

Using cross product prove that if $\vec{u} \times \vec{v} = \vec{0}$ and $\vec{u} \cdot \vec{v} = 0$ then $\vec{v} = \vec{0}$

I am asked to elaborate on the following proof: Let $\vec{u} \neq \vec{0}$. Prove that if $\vec{u} \times \vec{v} = \vec{0}$ and $\vec{u} \cdot \vec{v} = 0$ then $\vec{v} = \vec{0}$. My attempt ...
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4answers
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False proof: $\pi = 4$, but why?

Note: Over the course of this summer, I have taken both Geometry and Precalculus, and I am very excited to be taking Calculus 1 next year (Sophomore for me). In this question, I will use things that I ...
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1answer
42 views

Prove that locus of orthocentre is $x+y=0$

The base of a triangle is axis of $x$ and its other $2$ sides are given by the equations :$y = (1+α)\frac{x}{α} + (1+α)$ and $y = (1+β) \frac{x}{β} + (1+β)$.Prove that the locus of its orthocentre is ...
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1answer
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Pair of straight lines problem: Prove that $g (a_1+b_1)=g_1 (a+b) $

If the lines joining the origin and the point of intersection of the curves $ax^2+2hxy+by^2+2gx=0$ and $a_1x^2+2h_1xy+b_1y^2+2g_1x=0$ are mutually perpendicular then prove that $g (a_1+b_1)=g_1 (a+b) $...
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7answers
551 views

Get the equation of a circle when given 30 points [closed]

A similar question has been asked before on this site but that was of getting equation of circle using 3 points. I want my center to be more accurate So my question is how can i get the center of ...
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3answers
39 views

Verify if $\overrightarrow{w}$ is a linear combination of $\overrightarrow{u}$ and $\overrightarrow{v}$

I am given the following question: Let $\Vert \overrightarrow{u} \Vert = \Vert \overrightarrow{v} \Vert = \Vert \overrightarrow{w} \Vert = 1$ and $\overrightarrow{u} \cdot \overrightarrow{v} = \...
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1answer
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Show that, if $u,v,w$ are orthogonal two-by-two, then $S = \{ u , v , w\}$ forms a basis which is linearly independent

I am given the following question: Show that, if $u,v,w$ are orthogonal two-by-two, then $S = \{ u , v , w\}$ forms a basis which is linearly independent. My idea to tackle this problem is to ...
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1answer
19 views

intersection of a line (certain direction) and a circle

I need to calculate (previously) the point where a ball will touch the inside of a circle (for a game I'm developing). So I have two equations, one of the direction of the ball, and another of the ...
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2answers
742 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 ...
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1answer
36 views

Dimensions of bounding box for arbitrary circle sector

I need to determine the dimensions of bounding box for arbitrary circle sector as shown in the diagram below. Given: φ = Start angle in the range of 0 ~ 2π θ = Sweep angle in the range of 0 ~ 2π r =...
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2answers
43 views

Largest bifocal triangle in an ellipse

Ellipse $E$ has foci at $P$ and $Q$, and semi-major and semi-minor axes of length $a$ and $b$, respectively. Find the area of the largest triangle that can be (parttially) inscribed in ellipse $E$, ...
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2answers
41 views

Volume calculation with change of variables

I am trying to calculate the volume of a solid, given the equations of its bounding surfaces. It is a $3$-dimensional object, so the equations are in $x$, $y$ and $z$. In order to simplify the ...
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1answer
26 views

Analytical geometry - Finding the coordinates of point M

I've been practicing analytical geometry lately and I've come to a problem. I solved the problem a few times but I can't get the right result. Here is the math problem: Point M whose distance ...
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0answers
52 views

Congruent number $23$

The set of Congruent numbers are all the integer areas of rational sided right triangles. This means that if g is a Congruent number there exists some integer $n$ such that $g \cdot n^2$ is the ...
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1answer
52 views

Show that pair of straight lines $ax^{2}+2hxy+ay^{2}+2gx+2fy+c=0$… meet coordinate axes in concyclic points.Also find equation of

Show that pair of straight lines $ax^{2}+2hxy+ay^{2}+2gx+2fy+c=0$ meet coordinate axes in concyclic points. Also find equation of the circle through those cyclic points My Attempt Given equation ...
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1answer
18 views

Calculate equally-spaced and arranged center points on the edges of a circle based on its diameter and grid origin?

Question for a project: I have a circle that I know the diameter of (and therefore also the height/width dimensions of on a grid based on its top/left origin X/Y)... but I wanted to calculate equally ...
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1answer
62 views

Ellipse and parallel lines

Let's imagine that we have an ellipse described by the known equation $v^TAv=0$, (Link_1) where $v=[x \ y \ 1]^T$ (it can be a skew one in a general case). Then we have all possible parallel lines - ...
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1answer
37 views

graph of $z=2x+y$

The usual technique using traces where one variable is set to 0 does not seem to work here since I get all 0's and so where do the intersections meet? I looked in my calc. book and the technique ...
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4answers
100 views

Graph of the function $\cos(x)\cos(x+2)-\cos^2(x+1)$ will be?

Graph of the function $\cos(x)\cos(x+2)-\cos^2(x+1)$ will be? (A)A straight line (B)A parabola Give the corresponding equation too. Source:JEE 1997. Can ...
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0answers
38 views

Equations of the tangent planes to the sphere

Find the equations of the tangent planes to the sphere $x^2+y^2+z^2-10x+2y+26z-113=0$ which are parallel to the straight lines $\frac{x+5}{2}=\frac{y-1}{-3}=\frac{z+13}{2}$ and $\frac{x+7}{3}=\frac{y+...
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1answer
17 views

2D AABB count vector for not having collision after movement

I hope it is correct here, I feel like this question is more math related than programming. Table of Contents Introduction What is question about / problem description What way I figured out Other ...
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0answers
13 views

Condition for n points in the plane to determine a convex n gon

Suppose there are n points in the plane, labelled 1 through n, no three of which lie on a line. Suppose further that for every triple [i,j,k] with i< j < k that travelling from i to j to k is ...