# Tagged Questions

geometry assuming the parallel postulate of Euclid: in a plane, given a line and a point not on that line, there is exactly one line parallel to the given line through the given point.

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### What precisely is the difference between Euclidean Geometry, and non-Euclidean Geometry?

I was wondering, what it is precisely which defines the difference between Euclidean and non-Euclidean Geometry, in a few words/equations/diagrams? Would I be correct in understanding that non-...
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### What is the meaning of “integral point”?

While reading this paper (http://cowles.econ.yale.edu/P/cd/d04b/d0473.pdf) I encountered the concept of "integral point", used first in definition 5.1, on page 34. Does anybody know more details about ...
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### Megiddo's algorithm for lines of least weighted sum distance from a set of points

I came across the following problem: Given a set of n points (coordinate in 2d plane) within a rectangular space, find out a line ($ax+by=c$), from which the sum of the perpendicular distances of all ...
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### categorical description of the Minkowski sum of polytopes

Consider the category $\textbf{Poly}$ of polytopes, where the objects are convex hulls of finite subsets of $\mathbb{R}^d$ for arbitrary $d \in \mathbb{N}$ and where the morphisms are affine maps (i.e....
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### About a solid which satisfies $\sum_{i=1}^{n}x_i=0, |x_i|\le1\ (i=1,2,\cdots,n)$

For $n\ge 2\in\mathbb N$, let $S_n$ be the volume of a $(n-1)$ dimensional solid which satisfies $$\sum_{i=1}^{n}x_i=0, |x_i|\le1\ (i=1,2,\cdots,n).$$ Then, here is my question. Question : Can ...
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### Problem in understanding a proof there are five Platonic solids.

Thanks to several comments by Gerry Myerson, it is now clear that I wasn't clear, up to a state where I seriously confused myself. In a renewed attempt: Recently, I've been thinking about Platonic ...
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### How to find the distance between two planes?

The following show you the whole question. Find the distance d bewteen two planes \begin{eqnarray} \\C1:x+y+2z=4 \space \space~~~ \text{and}~~~ \space \space C2:3x+3y+6z=18.\\ \end{eqnarray} ...
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### Finding the “middle 2” of four lines

This may seem like an overly abstract problem, but it's the best generalization I could make of a specific problem I'm trying to tackle. This problem works in 2-dimensional Euclidean space. A ...
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### Characterization of Convex Polygons

John Lee's Axiomatic Geometry has an interesting characterization of convex and non-convex vertices for polygons. Let $P$ be a polygon. Consider a ray emanating from a vertex of $P$ which does not ...
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### Tiling an L-shape with “almost square”s

ABSTRACT: Define an "almost square" as a rectangles with aspect ratio in $[{1 \over 2},2]$. What is the minimal number of interior-disjoint almost-squares required to tile the following L-shape (where ...
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### How to find the points of tangency of a parabola using Calculus?

How can someone find the points of tangency of a parabola in this situation? I need to find two points of tangency so that the triangle formed by the two tangent lines at those points and the x axis ...
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### Is Euclid's Fourth Postulate Redundant?

Euclid's Elements start with five Postulates, including the fifth one, the famous Parallel Postulate. Less well, known however is the Postulate that forms the basis for motivation behind the fifth: ...
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### Which is the correct definition of stationary point for real-valued functions in Euclidean space?

Given a multivariable real-valued function $f$ whose first partials all exist (but which aren't all continuous) at $p$, it is possible that $f$ is not (totally) differentiable at $p$. But since the ...
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### Inequality in triangle

Let $ABC$ be a triangle and $M$ a point on side $BC$. Denote $\alpha=\angle BAM$, $\beta=\angle CAM$. Is the following inequality true? $$\sin \alpha \cdot (AM-AC)+\sin \beta \cdot (AM-AB) \leq 0.$$
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### Finding the radius of a circle

Given a point A that outside a circle so that $AT$ is tangent to the circle in point $T$ And $AC$ is a secant to that circle in points $B,C$. From points $B,C$ we build heights to $AT$ in ...
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### Proving a triangle is isoceles

In the graphic we have an isosceles triangle, and the problem is Calculate $\text{m}\angle BCD$ I added the point $E$ at distance $x$ from $C$ because it causes $DE=x$, after playing with ...
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### Which is the correct Hessian matrix (the standard matrix of a bilinear form)?

Please note the typo in the first entry: $\frac {\partial^2f} {\partial x_1\partial x_2}$ should instead be $\frac{\partial^2f} {\partial x_1\partial x_1}$. Also, this Hessian matrix need not be ...
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### Decomposable Families of Shapes

There are two types of golden triangles in the world, as shown in the following picture: Here $\varphi = \dfrac{1+\sqrt{5}}{2}$ denotes the golden ratio. Each of these golden triangles can be ...
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### Unit length vectors that sum to zero

Let's say we have a collection of $n$ vectors in $\mathbb{R}^2$ where $n$ is odd. Suppose each vector has unit length and that the sum of the vectors is zero. Is it necessarily true that the vectors ...
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### Show centers of squares formed by a parallelogram form a square.

A homework question I have been having some issue with - Given parallelogram $ABCD$, generate 4 squares from the sides of the $ABCD$. Given the 4 centers of the squares $W, X, Y, Z$ (formed by their ...
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### Projection on a hyperplan and a hypercube intersection

I need to project an array y onto a hyperspace defined by (a.x) = c where a is an array in R^N However, x needs to belong in the hypercube {0 <= x_i <= 1, for all i from 1 to n} Therefore from ...
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### cutting a cake without destroying the square toppings

There is a square cake. It contains N toppings - N disjoint axis-aligned squares. The toppings may have different sizes, and they do not necessarily cover the entire cake. I want to divide the cake ...
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### Are side lengths enough to find the ratio of the diagonals of a quadrilateral?

Is it possible to find the ratio of two diagonals of a quadrilateral when the length of all sides are given??
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### Number of Lines Passing Through a Given Point in the Plane

How can one prove that infinite number of lines pass through a given point in plane, using Euclid's axioms (or Hilbert's, if necessary)?
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### About the inscribed sphere and the exspheres of a $n$-dimensional simplex

Let us consider $n$-dimensional simplex $K$ in $n$-dimensional Euclidean space. Let $r_0$ be the radius of the inscribed sphere of $K$, and let be $r_1, r_2, \cdots, r_{n+1}$ be each radius of the ...
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### Online Math Open Contest 2 Problem 50

In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be ...
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### About the 'minimum triangle' which includes a convex bounded closed set

Question : Is the following true? "Letting $K$ be a convex bounded closed set on a plane, then there exists a triangle $M$, which includes $K$, such that $|M|\le 2|K|$. Here, $|M|,|K|$ is the area of ...
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### Proof with congruence of angles

I came across a proof exercise from my proof work-book that I am stuck on. The questions says: Suppose we have angle PQR with P, Q, and R non-collinear, and ray QS distinct from ray QR such that ...
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### Circles in Complex Planes

Points on the circle centre C and radius r are given by the equation $|Z-C|=r$ or $(Z-C)(\overline{Z}-\overline{C})=r^2$. Where $Z = x + iy$. When multiplied out, I understand that we have Z\...
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### Properties of a Square

So I have that squares A and B are congruent and one vertex of B is at the center of A. The question is what is the ratio of the shaded area to the area of square A. My question is if two square are ...
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### Prove that $l_1$ and $l_2$ are parallel if and only if $a_1b_2-a_2b_1=0$

For $b_1$ and $b_2$ non-zero, consider the lines $l_1=\{(x,y) \in \mathbb{R}^2 | a_1x + b_1y + c_1=0\}$ and $l_2=\{(x,y) \in \mathbb{R}^2 | a_2x + b_2y + c_1=0\}$. Assuming I only know Euclid's ...
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### Type of isometry

What type of isometry of $\mathbb{R}^3$ is the one given by sending $(x,y,z)$ to $(y,z,x)$ for each $x,y,z\in\mathbb{R}$? How can I find this isometry? Does anyone have a hint? My solution is that ...
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### How is the circle that fits beneath two adjacent circles related? [duplicate]

This is hard to search and probably easy to solve, but I keep finding articles about intersecting circles, and that is not what I'm after. I don't know what to tag this under, so if you know how to ...
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### Some property of half-planes in Euclidean and non-Euclidean geometry

Consider the Cartesian product of the set of real numbers $\mathbb{R}^2$ with the standard Euclidean metrics. An open half-plane is any set of pairs of real numbers $x$ and $y$ such that $ax+b> y$, ...
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### Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...