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

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2answers
53 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, ...
0
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1answer
101 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 ...
1
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0answers
38 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 ...
0
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1answer
85 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 ...
5
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1answer
85 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 + ...
0
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1answer
26 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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2answers
33 views

Finding the equation of a line without a given slope

Determine the equation of the line that contains the intersection of the lines $-4x+3y=-8$ and $-10x-4y=3$, and that has the same $y$ intercept as the line $$x=-\frac{2}{3y}-\frac{5}{4}$$
1
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1answer
50 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 ...
0
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1answer
38 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 ...
0
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2answers
113 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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0answers
53 views

Coordinate Geometry of the Circle

I have a question that I am completely stumped with, I just cannot figure what I need to do. The question is: Circle p has a center (q,w) and a radius r. Circle s touches circle a and b ...
1
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0answers
38 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 - ...
1
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2answers
37 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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2answers
93 views

Prove this trigonometry identity [closed]

Prove that $$\cot B=\frac{\tan{A} - \tan B}{1 + \tan A\tan B}.$$
0
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1answer
42 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?
1
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1answer
137 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 ...
3
votes
1answer
51 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 ...
37
votes
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 $n\rightarrow \infty$? ...
1
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1answer
86 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 ...
2
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1answer
19 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, ...
0
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1answer
33 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 ...
0
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0answers
33 views

Smooth compact subvariety from a germ of an embedding?

I am interested in the extent to which a germ of an embedding determines a subvariety given certain global hypotheses. My intuition is that the answer should be yes under more general conditions than ...
0
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1answer
31 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 ...
1
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1answer
56 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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0answers
32 views

Proving results about successive reflections of a point in vector geometry

If $Z$ is an arbitrary point in the plane and $$H_A:Z \mapsto 2A-Z$$ ie: $H_A$ denotes the reflection of a point $Z$ at $A$ Prove that for some point $Z$, $$H_A\circ H_B(Z) = 2 \vec{AB}$$ And ...
0
votes
2answers
120 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 ...
1
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1answer
42 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 ...
0
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1answer
33 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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1answer
37 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 ...
0
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1answer
122 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
83 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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4answers
82 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?
2
votes
2answers
197 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 ...
1
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1answer
39 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!
2
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1answer
36 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 ...
4
votes
2answers
72 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 ...
0
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1answer
75 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
35 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 ...
2
votes
2answers
102 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 ...
2
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1answer
119 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
48 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
57 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
88 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
70 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
39 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
63 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 ...
1
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1answer
98 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 ...
0
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0answers
30 views

Help solving a function

I have $2$ equations and $2$ angels that i need $V_{1}$ and $V_{2}$. I know The Point $(X_{m}, Y_{m})$ and the point $(X_{a}, Y_{a})$. I have one point $(X_{p}, Y_{p})$ that moves with the equation ...
2
votes
0answers
122 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 ...
0
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1answer
47 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.