# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $P = (2,4,-1)$?
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. ...