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

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2answers
353 views

I want to find 3 planes that each contain one and only one line from a set

The three lines intersect in the point $(1, 1, 1)$: $(1 - t, 1 + 2t, 1 + t)$, $(u, 2u - 1, 3u - 2)$, and $(v - 1, 2v - 3, 3 - v)$. How can I find three planes which also intersect in the point $(1, 1, ...
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1answer
406 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
223 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
669 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 ...
2
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1answer
395 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
539 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
357 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
527 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 ...
4
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1answer
265 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 ...
2
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1answer
3k 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
928 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
5k 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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2answers
170 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
147 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
302 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
5k 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
76 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
237 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
5k 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
10k 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
372 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
155 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
809 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
225 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
101 views

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
1k 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 ...
0
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2answers
488 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
199 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
2k 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
2k 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
105 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 ...
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0answers
361 views

Prove a very original version of Descartes's circle theorem

Prove: I define the radius of three mutually externally tangent to be $d,e,f$ respectively. The circle with radius $x$ is internally tangent to all three circles. Then $$ddeeff+ddeexx+ddffxx+eeffxx ...
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3answers
1k views

Find unknown coordinates of points

I hope it's enough understandable.
3
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1answer
118 views

Determinant and Measure

The determinant of the matrix of its vectors gives the measure of an $n$-dimensional parallelogram. For example, in $2$ dimensions, the area spanned by vectors $v$ and $w$ is \begin{array}{|cc|} v_1 ...
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2answers
300 views

Can more than four circles internally tangent or external tangent or combination of both each others at different points?

Is it true for infinite number of m, more than four, there exist m circles internally tangent or external tangent or combination of both each others(in this problem, i mean a circle must be tangent to ...
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1answer
385 views

closest point on a plane to another point in $\mathbb{R}^3$

Given $4$ points in $\mathbb{R}^3$: $A(0,2,4);B(-2,6,-2);C(2,-4,8);D(10,2,0)$, find the line equation $AK$ when $K$ is the projection of $D$ on the plane $ABC$. The first thing I did was find the ...
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1answer
158 views

Disk integration method to find volume of solid of revolution

I know that in a classic Cartesian coordinate system $xOy$, if I have a function $y = f(x)$ and I want to find the volume of the a solid of revolution around x-axis I can compute: $$V = \pi ...
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2answers
57 views

Are these planes?

Given the equation: y + z = 10 Can it be considered a plane? Why (not)? How do you correctly express planes which are normal onto one axis, for example a plane that lies completely vertical in space ...
0
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1answer
115 views

What is the formula of the following?

Let $S$ be the ellipsoid given by the formula $$ \frac{x^2}{a^2}+\frac{y^2}{b^2} +\frac{z^2}{c^2}=1$$ where $a \ge b \ge c > 0$ are fixed constants. What is the formula given by the set consisting ...
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1answer
38 views

Find a circumference with center on a line

I have a set of circumferences $$x^2 + y^2 + \alpha_1 x + \beta_1 y + \gamma_1 + k(x^2 + y^2 + \alpha_2 x + \beta_2 y + \gamma_2) = 0$$ $\alpha_1, \alpha_1, \beta_1, \beta_2, \gamma_1, \gamma_2$ ...
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3answers
3k views

Find a point on a line segment, located at a distance $d$ from one endpoint

Given points $A$ and $C$ in the plane, how do I find the point $B$ on the line segment between $A$ and $C$ that is located at a distance $d$ from $A$? Example: $$A = (0,3), \qquad C = (3,0), \qquad ...
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2answers
579 views

Plane from intersection line and point

The task: Determine the plane containing point $P( -5 , 2 , 3 )$ and going through the intersection line of the planes $2x + y + 5z = 31$ and $-4x + 5y + 4z = 50$ 1.: Intersect the two given planes, ...
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1answer
140 views

Equations in Analytic Geometry

There are many equations in Analytic Geometry like equation of a line, equation of a plane etc. My question: 1) Why equations instead of functions? 2) Why do equations almost always equal zero?
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1answer
164 views

Is there any way to give sense to a geometric/visual proof?

Suppose one is given the following visual proof that $$\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{1}{2^k} = 1$$ which is the following construction over $[0,1]\times[0,1]$ What this is ...
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1answer
294 views

What do the parameters skewX and skewY mean in the transform specified by Flash's motion XML?

Flash has the ability to export animations into a format they call motion XML. Its specification is here I am trying to write a python renderer for these animations using pyglet. I understand ...
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1answer
127 views

Directional cosines of a line.

Show that if the lines with the directional cosines $(l, m, 0)$ and $(p, 0, q)$ are perpendicular then either $m = \frac {1}{\sqrt{p^2 + q^2}}$ or $q = \frac {1}{\sqrt {l^2 + m^2}}$.
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1answer
79 views

One spot and distance known, Second spot unknown

I know the coordinates of points E and Q, so I know their euclidean distance L. I'm looking for the point W with coordinates (a,b) related to other known values?
4
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3answers
623 views

What is the tangent plane equation on the 3 spheres?

3 spheres are on $z=0$ plane and touch each other as shown in the picture. Coordinates of their centers are $O_1=(0,0,5),O_2=(0,y_2,3),O_3=(x_3,y_3,2)$. What is the tangent plane equation on 3 ...