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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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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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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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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72 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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59 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
43 views

Non-equivalent phrasings of Playfair's Axiom which are in use

For example on ProofWiki Playfair's Axiom is given as Exactly one straight line can be drawn through any point not on a given line parallel to the given straight line in a plane. but for example ...
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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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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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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
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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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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2answers
13k views

coordinate geometry: finding the ratio in which a line segment is divided by a line

The question is: Determine the ratio in which the line 3x + 4y - 9 = 0divides the line segment joining the points ...
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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 ...
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1answer
22 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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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 ...
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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, ...
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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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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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4answers
2k views

Hyperbolic critters studying Euclidean geometry

You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise. ...
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1answer
353 views

Is there a geometrical proof of the impossibility of squaring the circle?

The impossibility of certain constructions in Euclidean geometry, such as squaring the circle with straight-edge and compass is usually shown by using algebraic methods. I am wondering if there are ...
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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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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
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
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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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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How to find a point on a line closest to another given point?

Given the line $x+2=\frac{y+4}{2}=\frac{z-5}{-2}$ I want to find the closest point on this line to $(1,1,1)$ I suppose the details here don't matter but in general how is this done? We need a ...
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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
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Sum of distances from triangle vertices to interior point is less than perimeter?

Let $M$ be a point in the interior of triangle $ABC$ in the plane. Prove $AM+BM+CM<AB+BC+CA$. The above question was posed to someone I know who is taking high-school Euclidean geometry. I'm ...
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1answer
50 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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504 views

Intuition - The Shortest Curve Between Two Points is a Line

The references below aver that the following is not crudely trivial: The shortest curve between two points is a (straight) line. An elementary school teacher construed it as follows ...
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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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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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1answer
197 views

Average Degree of a Random Geometric Graph

A set of $N$ points are distributed randomly on a unit square with uniform distribution. Two points $\mathbf{p}_i$ and $\mathbf{p}_j$ are said to be connected if $\|\mathbf{p}_i - \mathbf{p}_j\| \leq ...
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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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19 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 ...
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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 ...
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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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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 ...
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2k views

Book with lots of geometry theorems

I want to study geometry and was looking for some book that has lots of theorems and covers almost all Euclidean geometry that is needed for High School and Maths Olympiads. Thanks.
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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 $B'_1, C'_1$ ...
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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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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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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. ...
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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?
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Is it possible to solve any Euclidean geometry problem using a computer?

By "problem", I mean a high-school type geometry problem. If no, is there other set of axioms that allows that? If yes, are there any software that does that? I did a search, but was not able to ...
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1answer
67 views

Extend isometry on some cube vertices to the entire cube

Let $K\subset V=\{-1,1\}^n$ be a set of vertices of the $n$-dimensional hypercube $D=[-1,+1]^n$ and let $f:K\to V$ be an isometry with respect to the Euclidean metric inherited from $\mathbb R^n.$ We ...
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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 ...