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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2
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38 views

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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0answers
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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14 views

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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1answer
68 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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4answers
58 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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2answers
74 views

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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0answers
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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1answer
71 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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1answer
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 ...
2
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1answer
61 views

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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1answer
21 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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4answers
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
18 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 ...
4
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1answer
107 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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1answer
31 views

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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1answer
41 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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2answers
39 views

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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2answers
39 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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1answer
40 views

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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0answers
36 views

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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1answer
66 views

Geometric proof for irrationality of $\pi$

Is there a geometric proof for irrationality of $\pi$? That would be neat.
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1answer
47 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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2answers
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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1answer
29 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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0answers
54 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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0answers
33 views

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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0answers
18 views

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 ...
4
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4answers
126 views

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, ...
2
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1answer
21 views

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 ...
0
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1answer
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 ...
2
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0answers
28 views

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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0answers
23 views

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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0answers
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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53 views

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. ...
2
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1answer
40 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 $B'_1, C'_1$ ...
4
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1answer
54 views

$\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could then the $\pi$ also be an integer?

$\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could there be a geometry where $\pi$ is a rational number or an integer?
4
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2answers
72 views

Prove that $u\cdot v = 1/4||u+v||^2 - 1/4||u-v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$

I need some help figuring out how to work through this problem. Prove that $ u \cdot v = 1/4 ||u + v||^2 - 1/4||u - v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$. Sorry, forgot to ...
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0answers
39 views

Common Intersection Point of ellipsoids

Suppose we have two ellipses in $2$-dimensions centered at the origin. It is easy to visualize that (unless one is contained in the other) they will have $4$ points of intersection. Is it known ...
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1answer
19 views

Finding $y_{A,i}$ and $y_{B,i}$ in this geometric relationship problem.

Finding $y_{A,i}$ and $y_{B,i}$ in this geometric relationship problem. I'm an high-speed aerodynamics student. I am studying a sweptback wing like in the figure below (in green). Notice that I ...
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2answers
40 views

Probably very basic Euclidean geometry; Why is the following expression valid for a point along a straight line?

I am looking at constructible points in abstract algebra, particularly in $\mathbb{C}$. Alongside a proof of a theorem, I came across this expression which I cannot work out how it's been derived. It ...
2
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1answer
39 views

Four Spheres Intersect Along Circles: Prove That Circles' Planes Are Either $\parallel$ to The Same Line, Or Have a Common Point

Problem: Let $\,A,\,B,\,C,\,D\,$ be four distinct spheres in a space. Suppose the spheres $A$ and $B$ intersect along a circle which belongs to some plane $P$, the spheres $B$ and $C$ intersect ...
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0answers
17 views

Finding closest vector for all rows in a matrix

I have two matrices 1. D ($m \times n$) and 2. C ($k \times n$). Typically, $m \approx 10^4, n \approx 100, k \approx 100 $. For each row r in D, I need to find the index of the row in C that's ...
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1answer
36 views

Partitioning a convex object without cutting existing convex subsets

$C$ is a convex object in the plane. $D_1,\dots,D_n$ are pairwise-disjoint convex subsets of $C$ such that $D_1\cup\dots\cup D_n \subsetneq C$, like this: Is it always possible to partition $C$ to ...
3
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1answer
53 views

Size of a point. [duplicate]

I know this may sound too simple or maybe too absurd to discuss, but I am having a hard time visualizing a point in space! In Euclid's Elements a 'Point' is defined as Something which has no part. ...
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0answers
21 views

A vector which is perpendicular to two vectors not in the same plane

Assume that I have two vectors $v_1, v_2$ which are not parallel and they don't lie the same plane. How to find a third vector $n$ perpendicular to $v_1$ and $v_2$? You could take the cross prosuct, ...
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1answer
48 views

Conjugating rotation by another rotation

If $g ∈ \mathrm{SO}(3)$ is the rotation about axis $p$ by angle $α$, and $h$ is a rotation mapping $p$ to another line $q$, then $g$ conjugated by $h$ is the rotation about $q$ by the same angle $α$. ...
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1answer
22 views

How to rotate in quaternions but for 2d version for arbitrary angle?

I am trying to understand the idea behind rotating in quaternions, but first I want to understand the math for 2d rotation. I saw some youtube videos, and I know that for 2D, a point in 2D can be ...
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0answers
25 views

A right hexagon and right pyramid

Does it possible to obtain a regular hexagon as a section of right pyramid with the base of the form of regular pentagon? O.Ganyushkin