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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Prove that a projector is continuous using Cauchy-Schwarz's inequality

Given a vector $a$ in an Euclidean Space with $a\cdot a = 1$ ($\cdot$ = scalar product), then $P(b) = (a \cdot b)a$ defines the orthogonal projection $P$ on vector $a$. How do you show that $P$ is ...
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35 views

Why the number of symmetry lines is equal to the number of sides/vertices of a regular polygon?

Considering the/a definition of a regular polygon from Wiki : In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all ...
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1answer
39 views

Sphere inversion radius

The following question came to me while programming some visualizations for Mobius Transformations as a pet project to learn Mathematica. Generally, the inverse of a sphere under sphere-inversion is ...
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47 views

Simultaneously a translation and central dilation

Find all maps which are simultaneously a translation and a central dilation? So if we have $X=(x,y)$ and we have the translation and central dilation, respectively, $\tau_A(X)=X+A$ ...
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31 views

Rotate a line between x1y1 and x2y2 by angle α then finding the end of the rotated line

I have two points, $x$1$y$1 and $x$2$y$2 on a coordinate grid indexed in matrix convention (y increases as you go down, x increases as you go right). I want to rotate an imaginary line between ...
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30 views

Congruence of triangles: SSA criteria

It is well known that this criteria does not work in general. I am trying to answer to the following question if two triangles have two sides and the angle NOT between them equal, they are either ...
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Putnam 2015 and Ravi Substitution

Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c$. Express $$\sum_{(a,b,c)\in T}\frac{2^a}{3^b5^c}$$ as a rational number in ...
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Vector equation intersect of xy-plane

Let $$L: \vec r(t)=<4-2t, -7+3t,8-13t>, \qquad t \in \mathbb{R}$$ Does this line ever intersect xy -plane? So, <4-t,-7+6t,3+2t> = < a , b, 0>, where a,b is an arbitrary constant. ...
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13 views

Rotating vector $[x_1,x_2,…,x_n]^T$ to a vector of the form $[y_1,0,…,0]^T$ on $\mathbb{R}^n$.

Assume $x=[x_1,x_2,...,x_n]^T$ a vector in $\mathbb{R}^n$. I am trying to rotate the vector in such a way that I get a vector of the form $y=[y_1,0,...,0]^T$ in $\mathbb{R}^n$. This is probably ...
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18 views

Ambiguous case in euclidian geometry

Is it possible to locate $4$(or more than $4$) points on a plane such that every point is at an equal distance from every other point?
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33 views

Problem of two scales in a corridor

There are two scales: one of length $z$ and the other of $y$. these scales is placed in a row so that their ends are each supported on opposite walls of the hall and that the scales intersect. Also ...
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Given AM as a median to triangle ABC, with EF parallel to BC, prove AK median to triangle AEF.

Hello, I'm studying with a group of friends going through a geometry study guide but we don't know any of the answers because we can't check our answers anywhere. If anyone can help or give a hint, ...
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33 views

Concurrence of four Newton lines

Let $ABC$ be a triangle, let line $L$ be a line in the plane, $L_N$ = Newton line of $(BC, CA, AB, L)$. Show that the Newton lines of $(L, L_N, AB, AC)$ ; $(L, L_N, BC, BA)$; $(L, L_N, CA, CB)$ and ...
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Why am I getting the wrong formula for the area of a dodecagon?

More likely than not, I'm just making a simple algebraic mistake, but I can't seem to find it and so I would like some help. Divide a (regular) dodecagon into $12$ congruent isosceles triangles with ...
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22 views

Prove transformation isn't a translation

I have these maps from the plane to itself where $X=(x,y)$: $f(X):=(y,-x)$ $g(X):=(x+2y,y)$ I need to compute $fg$ and $gf$ and show that none of these compositions are simply translations or that ...
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Is it possible to think solid as the orthogonal projection from 4D?

Get an orthogonal projection's area from this fomular $S=S^{\prime}\cos{\theta}$ when the angle between two planes is $\theta$, is a popular idea, so I wonder is it possible to get a volume of various ...
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26 views

Computing the set of fixed points of a map

Compute the set of fixed points of the following map: $f(X) := (y,-x)$ when $X=(x,y)$ So for this, do I just have to solve the system of equation such as: $x=y$ $-x=y$? Plugging the first into ...
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87 views

Given two adjacent sides of a rectangle are equivalent, prove that the quadrilateral is a square.

In Geometry class today, we were talking about quadrilaterals and the types of them. I was wondering that if, given a rectangle with two adjacent equal congruent sides, if that was enough to prove it ...
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64 views

I have a hard time understanding this simple theorem: “If two lines intersect, then exactly one plane contains the lines.”

I'm sorry if this is an extremely simple question, but I'm honestly having a hard time understanding a theorem in my geometry book. Here is the theorem: "If two lines intersect, then exactly one ...
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UnFlattening a 1/2 Triaxial Ellipsoid: Reconstructing a Squashed Tortoise

BACKSTORY: I have a flat tortoise. I need to figure out its original dimensions. I'm a paleontologist, and the site I'm working at has produced a [Hespertestudo crassiscutata], a giant tortoise ...
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28 views

Easiest way to prove a congruency

Let's take ABCD a convex quadrilateral, such that the diagonals AC and BD are perpendicular, and the angles ABC and CDA are equal. What is the easiest way to prove that the triangles ABC and CDA are ...
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73 views

Is wolframalpha wrong (Plotting inequalities)

I just wanted to plot a simple inequality: $$-x \geq 4$$ and wolframalpha gives me the following plot: But I think it should look like this: Am I correct? If so why is wolframalpha producing such ...
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38 views

Is the converse true too?

LDMJ is a circle centered at O. Point K, on DJ, bisects chord LM. DSJ is another circle drawn using DJ as diameter. If $\alpha = 90^0$, then KS = KL. This can be proved by applying “power of a ...
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Missile Guidance Course Correction

Background: I am controlling a simulated (programming) missile in a 2D space (no drag). The missile knows which direction it wants to go (the intended velocity vector direction). It is always ...
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23 views

Show that chordal metric is topologically equivalent to the Euclidean metric

Consider $$d(x,y)=\frac{2\|x-y\|}{(1+\|x\|^2)^{1/2}(1+\|y\|^2)^{1/2}},\hspace{5mm}x,y\in \mathbb{R}^n.$$ $d$ is a metric in $\mathbb{R}^n$ known as chordal metric. I want to show that this metric is ...
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49 views

Determining whether the shape is a rectangle

While solving a problem, I came across a little hump which is impeding a pure solution. If there is a quadrilateral ABCD where $\angle B = 90^\circ$ and $AD = BC$ and $\angle D = \angle C$, is it ...
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1answer
19 views

Bounds on the angles between four unit-length vectors in three dimensional Euclidean space

Let's consider four unit-length vectors $\mathbf{s}_i$, $i=1,2,3,4$, in three-dimensional Euclidean space. Let $\theta_{ij}$ be the angle between $\mathbf{s}_i$ and $\mathbf{s}_j$. Given the set of ...
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109 views

Is this really an open problem? Maximizing angle between $n$ vectors

It is well known that the trigonal planar molecule (with bond angle $\alpha=120^{\circ}$) and the famous tetrahedral (with bond angle $\alpha\approx 109.5^{\circ}$) maximizes the angle between the ...
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How to prove that a collection of epsilon balls generates a basis for topology?

Specific question: "Suppose $X$ is a three dimensional Euclidean space with the standard Euclidean metric. Let $Y$ be the subset defined by $Y=\{P_1$ s.t. $P_1=(a_1,b_1,c_1)$ and $c_1=0\}$ and use ...
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42 views

Maximize distance to closest vertex inside triangle

Question: Let $\Delta ABC$ be a triangle. For any point $P$ inside or on the boundary of triangle, define $d(P)=\min\{\overline{PA},\overline{PB},\overline{PC}\}$. Find the maximum of $d(P)$ (in ...
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Proving angles are supplementary in isosceles triangle

Let $ABC$ be a triangle with $AC=BC$, and let $P$ be a point inside $\triangle ABC$, satisfying $\angle PAB=\angle PBC$. If $M$ is the midpoint of $AB$, show that $\angle APM+\angle BPC=180^{\circ}$. ...
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40 views

Median points collinear: At least one outside triangle

Suppose that ABC is a triangle and that $A'\in l_{BC}$, $B'\in l_{AC}$, and that $C'\in l_{AB}$. Prove that if $A', B', C'$ are collinear, then at least one of these points must be outside of the ...
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Show that $\lvert x-x_1\rvert=c\cdot\lvert x-x_2\rvert$ describes an $n-1$ sphere for $0<c<1$ and a hyperplane for $c=1$

My attempt: For $c=1$, it is easy to visualize a 2D plane lying between $x_1$ and $x_2$ and simplifying the equation$\lvert x-x_1\rvert=\lvert x-x_2\rvert$ gives $$x\cdot(x_1-x_2)=\dfrac{(\lvert ...
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Equivalent norms on the Euclidean space $\mathbb{R}^{n}$

Show that if $F:\mathbb{R}^{n} \to [0,\infty)$ be a Minkowski norm on $\mathbb{R}^{n}$, then $$\lambda^{-1}|(y^{i})|\leq F(y)\leq \lambda |(y^{i})|,$$ where $y:=(y^{i})\in \mathbb{R}^{n}$, $|.|$ is ...
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67 views

Geometric proof for irrationality of $\pi$

Is there a geometric proof for irrationality of $\pi$? That would be neat.
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52 views

Geometric interpretation of Leibniz formula for $\pi$

We know $\pi=4(\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}....)$. I'm wondering, is there a geometric interpretation of this identity. Can we prove this identity by finding a different way to ...
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80 views

construct triangle given $b-c$, $r$ and $h_{b}$

As in title: the problem is to construct triangle given difference of sides $b$ and $c$, then in-circle radius $r$, and height $h_{b}$. The problem is from a set of problems exercising various ...
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31 views

Prove that if the altitude and median of a triangle form equal angles with sides then the triangle is right.

Problem statement: Prove that if the altitude and median drawn from the same vertex of a nonisosceles triangle lie inside the triangle and form equal angles with its sides, then this is a right ...
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56 views

Number of deltahedra as a function of the number of faces

How does the number of deltahedra (polyhedra with only equilateral triangles as faces) with no holes grow asymptotically as a function of the number of it's faces? If we have this as $N=g(F)$ for ...
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1answer
19 views

Metric Spaces Whose Diameter is Achieved at Every Point.

Suppose $(X,d)$ is a metric space with diameter $\sup \{ d(x,y) \colon x,y \in X\}=1$. Call the point $x \in X$ an edge point to mean that $d(x,y)=1$ for some $y \in X$. Call the metric space ...
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Let D denote a point on base AB, and let E denote a point on leg BC of an isosceles triangle ABC.

The triangles ABC, CDE, and BDE are all isosceles, and triangle BDE is similar to triangle ABC. Determine the angles of each triangle. Since ABC and BDE triangles are similar, their angles have to be ...
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Variant Lemoine's problem

You can see Lemoine's problem: Kiepert triangle: Let $ABC$ be a triangle, $BCA_0$, $CAB_0$, $ABC_0$ be three isosceles triangles constructed on the sides of $ABC$ with base angle $\alpha$. We called ...
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If $\sin A+\sin B+\sin C=\cos A+\cos B+\cos C=0$,

If $$\sin A+\sin B+\sin C=\cos A+\cos B+\cos C=0,$$ prove that $$\cos 3A+\cos 3B+\cos 3C=3\cos(A+B+C).$$ My solution: From the given, $$\cos^3A+\cos^3B+\cos^3C=3\cos A\cos B\cos C$$ Now, ...
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I have a convex hull with the facets in 3D. How do I compute the volume?

I have constructed a convex hull using Randomized Incremental Algorithm and I have the facets of the same. I need to compute the volume of this hull. Would some please share the algorithm for doing ...
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19 views

The rectangle-partition number and the number of horizontral edges

The rectangle-partition-number of a rectilinear polygon $P$ is the smallest number of pairwise-disjoint axis-parallel rectangles required to cover $P$. Some examples: (in the last example, $P$ is ...
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Proving that union of epsilon nets is an epsilon net

While reading a paper, I came across these definitions and claims: Definition: Given $p \in \mathbb R^d$, and $H$, a set of hyperplanes, let $$\text{Violate}_p(H) = \{h \in H: h \text{ is strictly ...
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Property of all k-dimesional shapes

A close friend showed me an intuitive pattern that given a k-dimensional convex polytope P, if one doubles every sidelength to generate a $P'$ it is possible to fit $2^k$ copies of $P$ within the ...
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24 views

Concurrence of lines through triangle midpoints [duplicate]

Suppose that points $A, B, C$ form a triangle and that $A'$ $\in$ $l_{BC}$ and $B'$ $\in$ $l_{AC}$ and $C'$ $\in$ $l_{AB}$ are such that the lines $l_{AA'}$, $l_{BB'}$, $l_{CC'}$ are concurrent at G. ...
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Concurrence through a triangle

Suppose that points A, B, C form a triangle and that A' $\in$ $l_{BC}$ and B' $\in$ $l_{AC}$ and C' $\in$ $l_{AB}$ are such that the lines $l_{AA'}$, $l_{BB'}$, $l_{CC'}$ are concurrent at G. ...
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41 views

Two new conics associated with two triangles inscribed a conic

1-Let $ABC$ and $A'B'C'$ be two triangle inscribed a conic. Let $P$ be a point on the conic. $PA$ meets $B'C'$ at $A_1$, define $B_1, C_1$ cyclically. $PA'$ meets $BC$ at $A'_1$ define $B'_1, C'_1$ ...