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

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Circle-Circle intersection coordinate system

Consider two points in the 2D Euclidean plane, the origin $0$ and $x$. One can define a co-ordinate system such that for any point $y$ in the plane, $y$ is parametrized by its distance from $0$, call ...
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1answer
103 views

Finding a coordinate with no intermediate variables

I want to know if it possible, using only the $+ - \div \times$ operators to solve a simple geometry problem. The questions is further complicated because I want to integrate it into a very restricted ...
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5answers
479 views

Can you prove why consecutive diagonal intersection points show decreasing fractions inside a rectangle?

When I was in third grade, I was playing with rectangles and diagonal lines, and discovered something very interesting with fractions. I've shown several math teachers and professors over the years, ...
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3answers
104 views

Smooth curve orthogonal to all hyperbolae $xy = a$ at points of intersection.

Suppose a smooth, connected curve $C$ in $R^2$ is orthogonal to all hyperbolae $xy = a$ whenever they coincide. I'd like to find the point(s) of intersection of $C$ with the hyperbola $xy = 16$ given ...
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1answer
191 views

A curve that intersects every plane in finitely but arbitrarily many points

Does there exist a piecewise smooth curve in $\mathbb{R}^3$ such that every plane intersects the curve at finitely many points and the number of intersection points can be arbitrary large? If the ...
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1answer
73 views

A shape that covers all box with certain side lengths

For a fixed $n$, what is the shape with the smallest volume, such that by rotation and translation, it can cover any $n$-box with dimension $b_1\times \ldots \times b_n$, where $b_1+\ldots+b_n=1$. I ...
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1answer
147 views

which of the following are homeomorphic?

well, I have forgotten how to identify ellipse, hyperbola,circle straightline from the general equation of conic, so is there any other way to identify these homeomorphic or not? a) B is an ellipse, ...
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2answers
447 views

Given a vertex and the base curve, how to find the equation of a cone [duplicate]

Possible Duplicate: Can any smooth planar curve which is closed, be a base for a 3 dimensional cone? Lets say a vertex V is given as $(\alpha ,\beta ,\gamma )$ and the base of the cone is ...
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1answer
120 views

Degree of Hessian surface invariant under linear transformations?

Given a surface $V(f) \subset \mathbb{P}^n$ for a homogeneous polynomial $f$ of degree $d$ on $\mathbb{P}^n$ and a linear transformation $g \in SL(n+1)$. Is the degree of the Hessian $H_f = V(\det ...
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2answers
184 views

Any video lectures on conventional analytical geometry?

Hi this question is kind of a natural offshoot to this question My topic covers very conventional topics like : Cartesian and Polar Coordinates in 3 Dim, second Degree eqns in 3 vars, reduction ...
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2answers
1k views

Find the magnitude of the acute angle between the lines $2y+3x=4$ and $x+y=5$.

Find the magnitude of the acute angle between the lines $2y+3x=4$ and $x+y=5$. I have no idea how to start the above equation. I try to draw the graph of $2y+3x=4$ and $x+y=5$ in the calculator ...
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1answer
99 views

Normal to the plane under the condition describes the cone

The plane $lx+my+nz=0$ moves in such a way that its intersection with the planes $ax+by+cz+d=0$ and $a'x + b'y + c'z+d'=0$ are perpendicular. Show that the normal to the plane through the origin ...
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4answers
277 views

Figure out if a fourth point resides within an angle created by three other points

If I have a point that is considered the origin and two lines that extend outwards infinitely to two other points, what is the best way to determine whether or not a fourth point resides on or within ...
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2answers
225 views

Using a single scalar equation to describe a line in space

A line in three-dimensional space may be described as an intesection of two planes, for example: $$\begin{align}x+y+z=0\tag{1}\\3x+7y=1\tag{2}\end{align}$$ This can be understood as two separate ...
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1answer
226 views

How to find the largest possible rectangle (by perimeter) on the following function?

It's been a while since I last tackled high-school math, and a friend asked me this question which I can't remember how to approach. I have the following: $ y = -x^2 + 5x $ Which produces an inverse ...
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2answers
534 views

Direction Cosines of the a line perpendicular to two lines

If $\alpha' ,\beta' ,\gamma'$ and $\alpha'' ,\beta'' ,\gamma'' $ are the direction angles of two lines, we have to find $\alpha ,\beta ,\gamma $ such that they are the direction angles of a third ...
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2answers
579 views

Finding relative coordinates in triangle

Known: $r$, coordinates of $A$, angle $BAC$=72° Task: find coordinates of B and C. So, I have 4 unknown parameters to compute, but only 3 equations. $r^2$=$(x_a-x_b)^2+(y_a-y_b)^2 $ $(2.49r)^2 = ...
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3answers
923 views

A good Open Source book on Analytic Geometry?

Hi my course specifically talks about : Cartesian and Polar Coordinates in 3 Dim, second Degree eqns in 3 vars, reduction to canonical forms, straight lines, shortest distance between 2 skew lines, ...
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1answer
355 views

Exercise review: perpendicular-to-plane line

Please, can you check the following execution is correct: Problem text I have a plane in affine space in $\Bbb R^4$ described by two following equations: \begin{Bmatrix}3x+y-z-q +1=0\\ ...
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4answers
135 views

Point on a line with the least distance from another point in $\mathbb{R}^3$

Consider the line $L$ defined by the following parametric equations $$x= 3+2t$$ $$y= 4+t$$ $$z=5-6t$$ Find the point $Q$ on $L$ that is closest to $(4,1,7)$. Note: I do not really remember the ...
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1answer
461 views

Coordinates of interception point Y with XY being the shortest distance of X to AB on sphere

How would one calculate the interception point $Y$ with $\overleftrightarrow{XY}$ being the shortest distance of $X$ to $\overleftrightarrow{AB}$? This answer to the question How to find the ...
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1answer
281 views

Prove that a conic section is symmetrical with respect to its principal axis.

A Calculus book that I'm self-studying is asking me to prove the following theorem about conic sections: A conic section is symmetrical with respect to its principal axis. Here is my attempt at ...
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2answers
517 views

Geometry IMO 1988

(IMO 1988/1) Consider two circles of radii $R$ and $r$ $(R > r)$ with the same center. Let $P$ be a fixed point on the smaller circle and $B$ a variable point on the larger circle. The line $BP$ ...
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1answer
278 views

Minimum sphere containing a tetrahedron

Is there an equation which would give me the radius of the smallest sphere containing a certain tetrahedron (no need to touch all vertices); given that I know the insphere, circumsphere radii and the ...
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1answer
451 views

Analytic proof for Circles of Apollonius

I'm looking for an analytic proof the statement for a Circle of Apollonius (I found a geometrical one already): If $\overline{AC}:\overline{BC}=s$, then $P \in k_s$. $s \in (0,1)$. $k_s$ is the ...
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1answer
226 views

Why it is sufficient to show $|f'(z)-1|<1$?

According to an article entitled "On the Univalency of Certain Analytic Functions" by Wang et al. (2006), we have to show that $|f'(z)-1|<1$ in order to find the radius of univalency for the class ...
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1answer
2k views

How to find a point after rotation?

Initially the position of the shape was in (100, 100). I am rotating (say 30 degrees) the shape as shown in the image below. I have found the starting point of the rotated object. Is there a formula ...
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2answers
540 views

Intersection of two lines using general form

How do I find the intersection of these two lines with their equations in general form. I don't want to graph them and I'm wondering if its possible with out converting them to gradient intercept ...
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4answers
3k views

Find the area of overlap of two triangles

Suppose we are given two triangles $ABC$ and $DEF$. We can assume nothing about them other than that they are in the same plane. The triangles may or may not overlap. I want to algorithmically ...
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1answer
107 views

An equation for all those points that have the same shortest distance to the same straight line in 3D space.

Can you form an equation for a ''pipe'' in 3D space? It means all those points P(x,y,z) that have the same shortest distance for the same straight line l. For example what would the pipe equation be ...
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1answer
133 views

What's the expected radius of a hypersphere

I would like to compute the expected radius of a hypersphere (dimension $N$) given these conditions: radius $R\in[R_{min}, R_{max}]$, radius is acquired by uniformly chosing point from ...
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1answer
226 views

Probability distribution of a coordinate of the random point on a hypersphere with given radius

If $(x_1,x_2,...,x_N)$ is a uniformly randomly chosen point on a hypersphere of a dimension $N$ with the radius $R$ (center in origin). What is the probability distribution of any coordinate? Done so ...
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2answers
3k views

Find an equation for the plane that contains the following line and passes through point P

How do you determine the plane which contains the line \begin{align} x & = -1 + 3t \\ y & = 5 + 2t \\ z & = 2 + t \end{align} and passes through the point $P = (2,4,-1)$?
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1answer
2k views

Locus of Z as cartesian equation

Could you please help with this locus problem? I think I am aiming for a cartesian equation in terms of $x$ and $y$ that may look like a circle equation e.g. $(x+a)^2 + (y+b)^2$ but I'm not sure. ...
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1answer
73 views

Vectors - For which value of t is the moving point A on $\vec{g}$ the closest to point B?

I'm having trouble finding a way to solve this particular problem: The point $A$ moves on $\vec{g}$ from point $J$ to $G$ and is dependent on the real parameter $t$: $\vec{g} = (-1/0/0) + ...
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1answer
192 views

Analytic Geometry

In our book of analytic geometry we have a title The canonical form of a line. It is the equation of a line passing through a point $p_1 := (x_1 , y_1,z_1)$ and parallel to a vector whose direction ...
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1answer
3k views

How to find the interior angle of an irregular pentagon or polygon?

I have 5 points and measures of sides of pentagon in 2D. Then how do i find interior angles of pentagon? Suppose $P_1,P_2,P_3,P_4,P_5$ are five points of Pentagon $P_1P_2P_3P_4P_5$. I know how to ...
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3answers
7k views

Finding out whether two line (segments) intersect

I need to know whether or not two line segments intersect. I thought the formula for that is y = mx + b but I don't think that will work for what I need, at least I think I need to first know whether ...
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3answers
197 views

Finding any point on a line if you know the slope and $y$-intercept.

I am wondering if there is a way to determine where a point is if I only know the slope and $y$-intercept. For example, say I am told that the line has a slope of $3$ and a $y$-intercept of $-3$. ...
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3answers
142 views

What equation intersects only once with $f(x)=\sqrt{1-(x-2)^2}$

Being $f(x)=\sqrt{1-(x-2)^2}$ I have to know what linear equation only touches the circle once(only one intersection), and passes by $P(0,0)$. So the linear equation must be $y=mx$ because $n=0$. I ...
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2answers
555 views

Finding point coordinates of a perpendicular

Given that I know the point coordinates of point $A$ and point $B$ on segment $AB$ and the expected length of a perpendicular segment $CA$ (perpendicular over $AB$), how do I calculate the point ...
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1answer
186 views

Which surface is formed by rotating a hyperbola around its asymptotes?

I don't know even what a type of surface will be. And what equation will be? The equation of hyperbola - $$ xy = l. $$ Now, let's $$ x = x'cos(\varphi ) - y'sin(\varphi ), y = x'sin(\varphi ) + ...
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0answers
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A function with the same slope as $b\sqrt{\frac{x^2}{a^2}-1}$ but not imaginary in [0,a]?

For some fixed $a,b \in \mathbb{R}$, $y = b\sqrt{\frac{x^2}{a^2}-1}$ is supposed to plot the boundary of an ellipse in $\left[0,a\right]$. I came up with that function but it has the defect that it ...
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2answers
890 views

How to calculate distance between point and object in 3d space

I have object in 3d space created from points $P_i(x, y, z)$ from which I can create triangles, and I need to calulate distance from point X to this object. I try to take 3 points from smallest ...
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2answers
357 views

Analytic Geometry in Space

Can someone help me solve the following two questions: 1) Find the distance between the lines: $$ L_1: \frac{x-1}{2} = \frac{y+3}{1} = \frac{z}{-1}$$ and $$\displaystyle L_2 : ...
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1answer
93 views

Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference?

Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference? And what is other branch of advanced analytic geometry called? in ...
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2answers
173 views

Find the parallels to a line which are tangent to an ellipse

Having the equation of a line, how can I find which of its parallels are tangent to an ellipse of equation $x^2 + 9y^2 = 1$? If the equation of the line is $y = mx + q$, I know that its parallels ...
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1answer
1k views

Finding a vertex of a triangle knowing the other two and its area

I have vertix A, vertix B and the area of a triangle, and I need to find the coordinates of vertex C, knowing that it's on the bisector between the first and the third sector of the Cartesian plane. ...
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1answer
1k views

Parametric equation for a line which lies on a plane

Struggling to begin answering the following question: Let $L$ be the line given by $x = 3-t, y= 2+t, z = -4+2t$. $L$ intersects the plane $3x-2y+z=1$ at the point $P = (3,2,-4)$. Find parametric ...
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2answers
97 views

Rectangle area and a curve

The diagonals of a rectangle are both 10 and intersect at (0,0). Calculate the area of this rectangle, knowing that all of its vertices belong to the curve $y=\frac{12}{x}$. At first I thought it ...