# Tagged Questions

For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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### parametric equations of folium of Descartes

We know the function of the folium of Descartes is $x^3+y^3=3axy$. The problem is to show that the folium of Descartes has parametric equations $x=\frac{3at}{1+t^3}$, $y=\frac{3at^2}{1+t^3}$ (this ...
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### Computing distances between hyperspheres and sides of a hypercube?

Suppose you are given the $n$ dimensional hypersphere: $$\left(x_1 - \frac{1}{2}\right)^2 + \left(x_2 - \frac{1}{2}\right)^2 +\ldots+ \left(x_n - \frac{1}{2}\right)^2 = \frac{n}{4}$$ And the ...
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### Fitting circle into an angle

I've been struggling with this for quite some time now, anyone could help me perhaps with this? Given an angle of an arbitrary degrees, and a circle with radius r. And imagine I would try to push the ...
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### Proof: Invariant angle measure - same result for any circle drawn.

Below I have quoted Wikipedia. I am particular interested in the statement: The value of $\theta$ thus defined is independent of the size of the circle: if the length of the radius is changed ...
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### 2D calculate position of a point relative to 4 known points

I have 4 known points (a square) in 2D space: A: {x:0, y:0} B: {x:100, y:0} C: {x:100, y:100} D: {x:0, y:100} Then I have a point inside the square. I don't know its location, but I do know the ...
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### find area of dark part

let us consider following picture we have following informations.we have circular sector,central angle is $90$,and in this sector there is inscribed small circle ,which touches arcs of sectors ...
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### Points in unit square

Let $n$ points be given in the unit square. How to prove or disprove: the points can be labeled $x_1,\ldots,x_n$ to satisfy the inequality $$\|x_1-x_2\|^2 +\|x_2-x_3\|^2+\cdots+\|x_n-x_1\| ^2 \le 4,$$ ...
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### Can the triangle inequlity extened to show the distance inequlity of a trapezium

$AB // CD$. What are the angle conditions (acute, obtuse or right angle) of $a,b,c,d$ to be satisfied the inequality $|AB+BC| > |CD|$? $AB,BC,CD$ are distances.
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### Term for changing properties in higher dimensions

Somewhat simple question, but it's the following. Consider D-volumes (that is, the equivalent volume measurement in D dimensions) of spheres of ever-higher dimensions. The percent of D-volume ...
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### how to find arc center when given two points and a radius

I am a math-illiterate, so I apologize if this doesn't make sense... I am working on trying to draw a custom interface using the iOS Core Graphics API. In a 2D space, I need to create a "rounded" ...
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### coordinates of icosahedron vertices with variable radius

I was looking on the wikipedia page about icosahedrons and it says that for edge length $a$ the radius of the circumscribed sphere around the icosahedron is given by $r = a \times sin(\frac{2\pi}{5})$....
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### largest empty sphere or rectangle

In N (~ 500) dimensions, I wish to find out the largest sphere or rectangle such that the sphere/rectangle does not contain already existing points. The entire set of points is bounded in an axis-...
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### Modification of the triangle inequality

We know from the triangle inequality that $X+Y \geq Z$, My question is under what conditions of $a,b,c$ (acute, obtuse or right angle) that $Z >X$ and $Z \geq Y$
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### Slices of a hypercube

Take the unit $d$-cube with vertices $\{0,1\}^d$, and restrict to the vertices that lie between (or within one of) a pair of parallel hyperplanes. These vertices form a graph whose edges are the edges ...
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### How can we prove that this triangle is Equilateral Triangle?

This is a problem which was sent to me by a friend , but i couldn't solve it , in particular , i don't have ideas for that . I hope you can help by hints or any thing . Here is the problem in the ...
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### Equal area rectangles in a piece of paper

If there is a paper that is split into $2$ sections (not to be assumed equal) with one section being laterally divided into $4$ equal subsections and the other $5$ (lateral means horizontal, or ...
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### index free proof of dot product of two wedge products

I am learning geometric algebra, and meet an identy of (edited according to Andrey's comments below) $$(a\wedge b)\cdot(c\wedge d) = (a \cdot d)(b\cdot c) - (a \cdot c)(b \cdot d)$$ as in wiki ...
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### Is it possible to divide a circle into $7$ equal “pizza slices” (using geometrical methods)?

Or is it possible to divide a circle into n equal "pizza slices" (I don't know how to call these parts, but I think you'll know what I mean), where n hasn't got a common divider with $360$? Or are the ...
Let $S^{n-1}$ be the unit sphere in $\mathbb{R}^{n}$. Let $\{e_1,\ldots,e_n\}$ be an orthonormal basis for $\mathbb{R}^{n}$. Let $\Sigma=\{e_1,-e_1,\ldots,e_n,-e_n\}$ be the set of $2n$ points ...