# 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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### Triangle whose height and sides are consecutive integers

This is probably a old puzzle,and maybe you have seen it somewhere else before.Imagine a special triangle. The height and the three sides of this triangle are 4 consecutive integers.Can you figure out ...
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### Eight queens problem, wondering about the non-unique solutions

I've done the code that generates all the solutions. But know I am suppose to filter out any redundant solutions based on symmetry and rotations. I have code for vertical symmetry, horizontal ...
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### Geometry (X : G), X isomorphic to G?

Let us have a geometry with a set X and a group G. What if X itself can be endowed with a group structure to be isomorphic to G? Can we gain something from this?
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### 2D transformation

I have a math problem for some code I am writing. I don't have much experience with 2D transformations, but I am sure there must be a straight-froward formula for my problem. I have illustrated it ...
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### Ten soldiers puzzle

This is a puzzle from one popular book called "The Man Who Counted: A Collection of Mathematical Adventures",author is Malba Tahan. How to arrange ten soldiers in five lines in such a way that each ...
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### Infinite sequence of nested, falling, colliding spheres

Imagine an infinite collection of nested, concentric spheres, of radius 1, $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, and so on. Suppose they are somehow suspended in space, fixed on their common ...
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### Given the area and height of a rectangle, what is the width of the base of a circular segment with the same height and area?

Given a rectangle of height $h$ and area $A$, what is the width $c$ of the chord at the base of a circular segment with the same height and area? I've made a diagram of the problem: My progress ...
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### How do I rotate a matrix transformation with a centered origin?

This is actually something I'm doing in Objective-C programming, but since it's very math-oriented I thought I'd post it here. I was reading up on linear transformations: ...
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### What's the simplest algorithm for resizing an object inside a rectangle so that it's as large as possible?

This is a simple enough problem that I could just cover all corner cases, but I was wondering if there was an elegant way to do this. Here is the starting point. It finds out which side of the image ...
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### Finding a homeomorphism guaranteed by Schoenflies Theorem

Assume I have a Jordan curve $C \subset \mathbb{R}^2$. Then by Schoenflies Theorem there exists a homeomorphism $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $f(C)$ is the unit circle. Is ...
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### An intuitive proof for one of the fundamental property of a parallelogram

"The sum of the squares of the diagonals is equal to the sum of the squares of the four sides of a parallelogram." I find this property very useful while solving different problems on ...
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### Is it possible to deduce a model for hyperbolic geometry from a synthetic set of axioms a la Euclid/Hilbert/Tarski?

Motivation I learned from Emil Artin's book Geometric Algebra that the standard incidence axioms of affine geometry (two points determine a unique line, parallel postulate, no three collinear points ...
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### Finding the intersection point of many lines in 3D (point closest to all lines)

I have many lines (let's say 4) which are supposed to be intersected. (Please consider lines are represented as a pair of points). I want to find the point in space which minimizes the sum of the ...
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### Sum of angles | Tetrahedron

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°. Any help on getting me started out here? I am not able to get any idea as ...
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### Rotation of a vector distribution to align with a normal vector

I generate a distribution of random points on a unit hemisphere whose pole is on the positive z-axis (the base lies in the x-y plane). Each point represents a directional vector $v$ in which a ray ...
1k views

### Best way to find the Coordinates of a Point on a Line-Segment a specified Distance Away from another Point

I have 4 points: $Q, R, S, T$. I know the following Coordinates for $R$, $T$, and $S$; Length of $\overline{RQ}$ That segment $\overline{RT} < \overline{RQ} < \overline{RS}$; I need to ...
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### algebraic way to compute intersection of disks

Is there a pure algebraic way to calculate intersection of two disks (extended to spheres, ellipses)?
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### A Sphere Containing Points of Pairwise Equal Distance

Suppose one has $m$ points in $\mathbb{R}^n$ with the property that the distance between any two of them is some fixed constant $d$. Is it true that there is a sphere (living in $\mathbb{R}^n$) ...
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### Orthogonal mapping $f$ which preserves angle between $x$ and $f(x)$

Let $f: \mathbf{R}^n \rightarrow \mathbf{R}^n$ be a linear orthogonal mapping such that $\displaystyle\frac{\langle f(x), x\rangle}{\|fx\| \|x\|}=\cos \phi$, where $\phi \in [0, 2 \pi)$. Are there ...
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### Solids produced from finite constructive solid geometry operations

Constructive Solid Geometry is a way of describing/building up solid objects from simpler primitive objects. Let's assume you can perform affine transformations on objects, along with the CSG set ...
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### Given a width, height and angle of a rectangle, and an allowed final size, determine how large or small it must be to fit into the area

In other words, if I had a rectangle of $10\times 10$ and an angle of $45$, and the allowed area was $100\times 100$, the rectangle would be about $70\times 70$. The allowed area is $100\times 100$ ...
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### What is the most accurate method to get intersection point in 3D?

I have been given 3D point data, belonging to different planar segments. Points are not exactly laid on the planes so that I have fitted best planes using least square solutions. Now, I want to find ...
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### How do I apply a speed to make a point travel along a line between two points

I have point $A$ which is traveling towards point $B$. Both points have $x,y,z$ coordinates. Point $A$ has a speed. For a given time period how much would I add to the $x,y,z$ coordinates of $A$ in ...
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### How to find a rectangle which is formed from the lines?

I have Cartesian coordinates $A$ and $B$. Line $AB$ is the axis (center) of the rectangle. And I have $H$ (height). I need Cartesian coordinates blue rectangle.
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### Name for this triangle centre

Given a triangle I draw circles around each vertex. I chose the radii of these circles so that they are all mutually tangent. There is only one way to do this. I extend these tangents. They concur at ...
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### Get Rotation in degrees (0-360) from a rotated angle?

I have a rectangle that is facing up. ($0^\circ$) I'm getting a number bettween $-1000$ to $1000$ or even more, and this number is the angle that is rotating the rectangle. How can I know the ...
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### Calculating points on a plane

In the example picture below, I know the points $A$, $B$, $C$ & $D$. How would I go about calculating $x$, $y$, $z$ & $w$ and $O$, but as points on the actual plane itself (e.g. treating $D$ ...