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

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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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1answer
143 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 ...
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1answer
94 views

Prove that $\|a\|+\|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.

Prove that $\|a\| + \|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane. Gentle hints only, please! I know that attempting to decompose R.H.S. into $$\alpha a + \beta b + ...
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1answer
28 views

There exist two points $P$ and $Q$ in $A$

I would appreciate if somebody could help me with the following problem Q: Let $A$ be a set of $n$ distinct points in $\mathbb{R}^2$. Prove that there exist two points $P$ and $Q$ in $A$ such that ...
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1answer
72 views

Product of gradients of x=0 and y=0

A friend asked me this question: The product of the gradient of any two lines perpendicular to each other is $-1$. Now, the lines $x=0$ and $y=0$ are perpendicular to each other. If you take the ...
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1answer
40 views

Is it valid to apply an operation to coordinates on a graph? Ex: $2(a,b) = (2a, 2b)$?

As the title says, is it valid to do something like $2(a,b)$ where $(a,b)$ are points on a graph, such that $(a,b)$ becomes $(2a,2b)$ ? or is this not valid because coordinates cannot be changed using ...
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2answers
125 views

Coordinate Geometry Triangle

ABC is a triangle. BB$_1$ and CC$_1$ are angle bisectors of B and C respectively. E,F are feet of perpendiculars from A on BB$_1$ and CC$_1$ respectively. Suppose D is point at which incircle of ABC ...
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1answer
71 views

shortest distance between two cones in 3-dim space

How can I find the shortest distance between two cones in 3-dim space? cone 1: apex - $(x_{0}, y_{0}, z_{0})$ angle - $\alpha_{0}$ base circle - $(cx_{0}, cy_{0}, cz_{0}, r_{0})$ cone 2: apex - ...
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2answers
42 views

How to prove $\frac{y^2-x^2}{x+y+1}=\pm1$ is a hyperbola?

How to prove $\frac{y^2-x^2}{x+y+1}=\pm1$ is a hyperbola, knowing the canonical form is $\frac{y^2}{a^2}-\frac{x^2}{b^2}=\pm1$ where $a$ and $b$ are constants? Thanks !
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1answer
48 views

$3x+3y-1,4x^2+y-5,4x+2y$ are sides of an equilateral triangle

I am completely lost in this one $3x+3y-1,4x^2+y-5,4x+2y$ are sides of an equilateral triangle, its area is closest to the which integer?
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1answer
215 views

Triangle - coordinate geometry problem

Let ABC be a triangle. Let BE and CF be internal angle bisectors of B and C respectively with E on AC and F on AB. Suppose X is a point on the segment CF such that AX is perpendicular to CF; and Y is ...
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1answer
64 views

Why is this an ellipse?

On a textbook, I've arrived at the following function: $\displaystyle \phi(z)=\log{\frac{|z-\sqrt{(z²-1})|}{2}}$ and it says that the formula has a simple interpretation: the level curves of ...
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9answers
3k views

What is this beauty curve?

Consider the following shape which is produced by dividing the line between $0$ and $1$ on $x$ and $y$ axes into $n=16$ parts. Question 1: What is the curve $f$ when $n\rightarrow \infty$? ...
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1answer
198 views

Coordinate Geometry Oblique Coordinates Problem

This is a elementary geometry problem which I have tried to solve using coordinate geometry but it is resulting in an impossible and impractical result. Maybe I have some misconceptions with oblique ...
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1answer
26 views

Sequence of coordinates on a polygon

If we have all coordinates of the vertices of an arbitrary polyhedron, is it possible to determine on what faces they are and in what order? Actually, I already know the first part of the question, ...
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1answer
82 views

A question of straight lines

If the straight lines $x+y-2=0$, $2x-y+1=0$ and $px+qy-r=0$ are concurrent, then what is the slope of the member of family of lines $2px+3qy+4r=0$ which is farthest from origin? I wrote the ...
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1answer
34 views

Given $\vec{A_1}(1,2), \vec{A_2}(2,4), \vec{A_3}(3,b).$ find $b$ so that triangle $\triangle{A_1A_2A_3}$ will be a right-angled triangle

Given $\vec{A_1}(1,2), \vec{A_2}(2,4), \vec{A_3}(3,b).$ find $b$ so that triangle $\triangle{A_1A_2A_3}$ will be a right-angled triangle. I know that in order that $\triangle{A_1A_2A_3}$ will be ...
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1answer
127 views

Connecting Coordinate Geometry and Plane Geometry

What is it that allows us to take theorems proven in Euclidean geometry (i.e. with Euclid's five postulates or Hilbert's Axioms) and then apply them outside of Euclidean geometry. For example in ...
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2answers
391 views

How to find out if a point lie in rectangle?

I have a rectangle in $2D$ space which is determined by $2$ points (each in opposite vertice) $p_1(x,y)$ and $p_2(x,y)$ . How can I find out numerically if a other point $p(x,y)$ is lying inside plane ...
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1answer
49 views

Intersection of curve and line

This is a question which I want to solve, taken from this sample question paper for an exam I'm appearing for tommorow: If a line, parallel to, but not identical with, x- axis cuts the graph of ...
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1answer
49 views

Fixed points through a general circle.

The circle $C: x^2 + y^2 + kx + (1+k)y - (k+1)=0$ passes through two fixed points for every real number $k$. Find $(i)$ co-ordinates of these two points and $(ii)$ the minimum value of the radius.
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53 views

Can we express these sets as Cartesian products of two subsets of $\mathbf{R}$?

Let sets $A$ and $B$ be given as follows: $$A := \{ (x,y) \in \mathbf{R}^2 | \ \ x < y \ \ \} $$ and $$B := \{ (x,y) \in \mathbf{R}^2 |\ \ x^2 + y^2 < 1 \ \ \}.$$ Can we express $A$ or $B$ as ...
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1answer
211 views

how to find the barycentric coordinates of the orthocenter

$A = (0,0),B = (4,0),C = (1,2)$ How can I find the barycentric coordinates of the orthocenter of $\triangle ABC$?
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2answers
143 views

Find the equation defining a perpendicular bisector

Hello fellows, I've not had much time to post questions, but I post this one because while in my Maths lesson, I became annoyed by solving the same thing over and over again, when a good ...
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5answers
173 views

If we are given a circle and its equation and a point which lies on it..can we find the diametrical opposite point?

If we are given a circle and its equation and a point which lies on it.. Can we find the diametrical opposite point?
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2answers
646 views

Is it true that all of the euclidean geometry problem in the IMO(international mathematical olympiad) could all be solve by the analytical geometry?

Is it true that all of the euclidean geometry problem in the IMO(international mathematical olympiad) or even generalize to say that all the plane geometry problem and 3d-geometry could be solve by ...
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2answers
57 views

Proving $proj_{proj_{\vec u} \vec v} \vec v=proj_{\vec u} \vec v$

Can anyone show me how to prove: $proj_{proj_{\vec u} \vec v} \vec v=proj_{\vec u} \vec v$? I got confused trying to prove it (not geometrically)... Thanks in advance!
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1answer
58 views

Bouncing of a ball from circular boundary

Lets say a ball with xspeed: 14, yspeed: 16 hits the circular edge at xposition:626 yposition:382 like on the below picture : It needs to bounce properly, to get the right bounce and new ball ...
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2answers
137 views

Question straight from the SAT

If a coordinate system is devised so that the positive y-axis makes an angle of 60 degrees with the positive x-axis, what is the distance between the points with coordinates (4,-3) and (5,1)? I'm ...
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1answer
134 views

Proving a vector equality in a triangle without using Thales' theorem.

Problem Let $\text{ABC}$ be a triangle, and $\text{M}$ and $\text{N}$ are points where: $\vec{\text{AM}}=\frac{1}{3}\vec{\text{AB}}$ and $\vec{\text{AN}}=\frac{1}{2}\vec{\text{AB}}$ and ...
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0answers
56 views

Question on a statement about analytic variety irreducible at $0$.

I am trying to understand this statement, it is in "Principles of Algebraic Geometry" by Philip Griffiths and Joe Harris. In page 13, third point, they are trying to prove that an analytic variety ...
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2answers
110 views

Prove that $\text{(BE)}\|\text{(JF)}$ using vectors.

Problem Let $\text{ABC}$ be a triangle and let $\text{I}$ , $\text{J}$ and $\text{K}$ be points such that: $\vec{\text{BI}}=\frac{1}{2}\vec{\text{IC}}$, $\vec{\text{AJ}}=2\vec{\text{JB}}$ and ...
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1answer
341 views

Geometric Interpretation of Jacobi identity for cross product

Is there a geometric "reason" for the Jacobi identity for cross products? Some geometric equality of some area ...? All proofs I know work by some form of linear algebra (or use the interpretation as ...
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1answer
50 views

square formed by the quadratic equations.

Question:A Square is Formed By The Straight Lines $x^2-8 x+12 = 0$ And $ y^2-14y+45 = 0$. What are the coordinates? How do I solve it? Providing a basic intiution will do the job. Also the graphs of ...
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0answers
66 views

Three reflection theorem in the context geometry on the sphere

Recently, I study geometry on the sphere in Patrick Ryan's Geometry book. A line on the unit sphere $S^2$ is defined as \begin{equation} l=\{x\in S^2: <x,z>=0\} \end{equation} for some point ...
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4answers
286 views

Proving using vectors, that if a median is also a height, then the triangle is isosceles.

Proving using vectors, that if a median is also a height, then the triangle is isosceles. *Better wording would be very helpful. Thanks in advance for any help.
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1answer
74 views

Problem with vector calculation.

Problem Let $\text{ABC}$ be a triangle and let $\text{A'}$ , $\text{B'}$ and $\text{C'}$ be respectively the center of $\text{[BC]}$ , $\text{[AC]}$ and $\text{[AB]}$. Prove that: ...
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3answers
45 views

Some basic questions about vectors

I've got two quite basic questions about vectors. I'm sorry if it isn't right to put two questions at the same thread. I'm quite confused about the technique of solving such problems. Let $\vec ...
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0answers
66 views

Probability of a triangle in a circle [duplicate]

I'm confused on my calculations on analytic geometry with probability. Things I learned on these were messed up since I was a newbie on these subjects. Here's my problem: Three points are chosen ...
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1answer
178 views

How to find the points of tangency of a parabola using Calculus?

How can someone find the points of tangency of a parabola in this situation? I need to find two points of tangency so that the triangle formed by the two tangent lines at those points and the x axis ...
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2answers
266 views

The locus of points that are equidistant from lines $y = x+3$ and $y = x+7$

How can I find the locus of point $P (x,y)$ that moves so that it is equidistant from lines $y = x+3$ and $y = x+7$? I take any point on the first line to be $M (x,x+3)$ and second line to be $S ...
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1answer
283 views

Ellipses given focus and two points

I would like to find all ellipses which contain 2 given points and has one focus at origin (zero). All in 2D plane. There are several possible approaches but I'm not sure which is the best - both ...
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1answer
116 views

equation of a perpendicular bisector [closed]

A diagram shown has point A( -2 , 4 ) , B ( 6, 2 ), C (-4,-4) find the equation of the line perpendicular to BC and passing through the midpoint of BC (M). Give answers in general form.
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1answer
32 views

Bisector equation

I have the coordinates of the triangle vertices (A(x1,y1),B(x2,y2),C(x3,y3)) . Could I write the bisector equation taken from the top of A (AM, M(x,y))? Thanks in advance.
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1answer
238 views

Why are two definitions of ellipses equivalent?

In classical geometry an ellipse is usually defined as the locus of points in the plane such that the distances from each point to the two foci have a given sum. When we speak of an ellipse ...
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1answer
1k views

Intersection of a plane with an infinite right circular cylinder by means of coordinates

So, I started studying analytic geometry and I must say I'm finding it much harder than "classic" geometry, because of the equations without help from diagrams... Still, I wanted to see how to use it ...
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1answer
65 views

How to calculate new position of a rectangle after translation and rotation?

I have a rectangle - lets say 100 long by 75 high. Origin been bottom left corner. I move the rectangle up and across by 10 and rotate by 3 degrees from centre of part. How do I calculate the new ...
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1answer
66 views

Family of curves $x^n+y^n=a^n$ as $n$ goes from $1$ to $\infty$ (integers) and from $1$ down to $0$

Take values of $n$ from 1 to $\infty$ in steps of $1$. Prove that in the limit it will be a square of side $a$. Take value of $n$ from $1$ to $0$ (fractions). In the limit as $n$ approaches $0$, prove ...
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3answers
223 views

Proof of Aristarchus' Inequality

Does anyone know how to prove that if $0<\alpha<\beta<\frac{\pi}{2}$ then $\frac{\sin\alpha}{\alpha}>\frac{\sin\beta}{\beta}$. Any methods/techniques may be used.
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2answers
44 views

How to do this problem with vectors?

A plane flies in a direction NW at an airspeed of $141$ km/hr. If the wind at the plane's cruise altitude is blowing with a speed of $100$ km/hr directly from the north, what is the plane's actual ...