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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Question about inequality in linear algebra

$V$ is inner product space. $u, v \in V$ are two orthogonal vectors. Prove that $\|v-u\| \geq \|v\|$. Because $\|v-u\|, \|v\| \geq 0$ it's enough to prove that $||v-u||^2 \geq \|v\|^2$. ...
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Find Cartesian and Vector equations of the plane perpendicular to (1, 0, -2) and containing the point (1, -1, -3)

I'll work through my current progress until I reach the bit where I get stuck. We are given $$n = (1, 0, -2)$$ Thus Cartesian form will be $$x - 2z = r\cdot n$$ Now, $$r\cdot n = (1, -1, -3)\cdot ...
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32 views

Two perfect squares in a right triangle

Prove that there is no integer sided right triangle in which the lengths of two sides are simultaneously perfect squares
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89 views

Geometry: Find angle x in triangle

I have not been able to find a euclidean geometry solution to this, but any other solutions are also appreciated. Let ABC be a triangle with AB=CD and angles as marked in the diagram. Find the ...
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0answers
20 views

Orthogonal spaces, span-formula

I read in a book this formula: ($v_i$ are vectors of an euclidean vector space, each one $\neq$ 0) $(\cap v_i ^\bot )^\bot = \sum v_i^{\bot \bot}$, The intersection and the sum are build over a ...
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52 views

Geometry Homework [closed]

A quadrilateral $ABCD$ is inscribed in a circle of center $O$. Chords $AB$ and $CD$ intersect at $M$ such that $AM=6, MB=4, MD=3$ and $MC=8$. The tangents at $A$ and $B$ to the circle meet in ...
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1answer
38 views

A geometry homework question

Let $A,B,C$ and $D$ be $4$ points in the plane such that any combination of three or more of them are non-collinear. Let $[AB]$ and $[CD]$ intersect at $M$. Suppose that: $AM = 6$, $MB = 4$ and $MD = ...
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1answer
19 views

Travelling Sales Man problem

I am studying the Travelling Sales Man problem: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and ...
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2answers
21 views

How to prove triangle inequality for euclidean norm on complex number?

We were asked to show that when: $\displaystyle \Vert Z\Vert = \left(\sum_{k=1}^{n} (x_k+iy_k)(x_k-iy_k)\right)^{1/2}$ that $\Vert Z+W\Vert \leq \Vert Z\Vert+\Vert W\Vert$ whenever $Z$ and $W$ are ...
4
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1answer
57 views

Is the following set (path) connected?

This is a homework question. $d,n\ge 2$. Let $L=\{(x_1,...,x_n)\in (\mathbb{R}^d)^n: x_i\in \mathbb{R}^d, x_i\ne x_j \forall i\ne j\}$. I tend to think it is not path connected because if you ...
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1answer
22 views

Position of a point on a line segent relative to the segent's length

I would like to ask for help with clarifying the following formula for calculation of relative position of a point on a line segment with respect to the line segment's length in two-dimensional ...
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2answers
62 views

How to draw an $405^\circ$ angle?

In a math test a question was to draw a $405^\circ$ angle. Is it formally correct to say draw an angle as I think that in geometry, an angle has just some formal definition. So what is the connection ...
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3answers
83 views

From Hartshorne's Geometry: Euclid and Beyond: contruct and inscribed equilateral triangle in a given circle

I haven't found a propert solution for this problem: (4.3) Given a circle, but not given its center, construct an inscribed equilater triangle in as few steps as possible. I managed to ...
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0answers
14 views

$\text{diam}(\Omega)$ is $\geq$ to at least one side of the minimal rectangular box containing $\Omega$?

For $\Omega\subset\mathbb{R}^n$ open and bounded, is it always the case that $\text{diam}(\Omega)$ is greater or equal to at least one side of the minimal rectangular box containing $\Omega$ ? Added : ...
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1answer
33 views

Euler charcteristic of the intersection of hyperplanes with a pointed cone

Consider a pointed closed convex polyhedral cone $C$ in $\mathbb R^n$. Let $S_1,..,S_n$ be hyperplanes in $\mathbb R^n$. Consider $S=C \cap \left(\bigcap\limits_{i=1}^n S_i\right)$. If $S$ is ...
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2answers
41 views

elementary geometry/algebra

I was looking at my geometry chapter summary on similar triangles, and I was a little confused with the result. I'm really tired right now and I am having difficulty leafing through the chapter to ...
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32 views

The formula of Eclidean distance to a hyperplane.

I have a hyperplane eqution H: "$X - Y = 0$" where $X, Y \in R^{n\times m}$. Could you tell me how to deduce the smallest Euclidean distance formula for any point ($X_0,Y_0$) to H ?
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1answer
34 views

How can I move a point along a line in 3D space to reach a target dot product with a fixed reference point?

Suppose a point in 3D space, Q. For any other point x in that space, Let Q(x) be the unit vector pointing from x towards Q. I also have a line L in 3D space, and a point on this line P. L = {P + ...
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0answers
28 views

Relating the incenters of the original and medial triangles.

Let I be the incenter of △ABC. If I is also the incenter of the medial triangle of △ABC, show that △ABC must be equilateral. I'm thinking a place to start would be to show the distance between AC and ...
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2answers
36 views

On some propreties of orthogonal complements

In my book the following propositions on orthogonal complements are given without any proof. However, I cannot figure out how to prove them, even though they must follow directly from the definition ...
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2answers
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The $ABCD$ paralelograms sides are $AB,BC,CD,DA$. On these line segments there are points in the same order: $X,Y,Z,V$.

The $ABCD$ paralelograms sides are $AB,BC,CD,DA$. On these line segments there are points in the same order: $X,Y,Z,V$. We know, that: $$\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=\frac{DV}{VA}=k$$ ...
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0answers
23 views

Huzita Axiom 6 - Computing the Origami Trisection of an Angle

The Galois theory proof of the improssiblity of angle trisection rests on studying the triple angle formula $\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta$. Ruler and compass numbers can only be ...
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1answer
7 views

Orthogonal matrix

I am given that the vectors $x$ and $x'$ have the same Euclidean length and $Qx=x'$ where $Q=I-\frac{2uu^T}{\|u\|^2}$ and $u=x-x'$. I need to show that $Q$ is orthogonal but I don't know how to do ...
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1answer
61 views

Rigorous books on geometry

I am looking for a rigorous book on both 2d and 3d euclidean geometry, and also how analytic geometry can be developed from synthetic geometry. I haven't really found such a book yet. I would be very ...
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2answers
48 views

Circles question on proof

It is given that a, b, and c are the sides of a triangle and c is the hypotenuse. There is an incircle inside the triangle with radius = r. We need to prove that $r=\dfrac{a+b-c}{2}$ Image: My ...
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4answers
103 views

Term for similarity transformation which is not a translation

What's the best (i.e. most concise) term to refer to an orientation-preserving similarity transformation which is not a translation? Here are some descriptions I could think of, but all of them feel ...
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2answers
46 views

Problem in proving that the locus of all points S is a circle.

Given is a circle with midpoint $M$ and a chord $AB$ on this circle. $S$ is the intersection of the altitude from $M$ to $AB$. Prove that the locus of all points $S$ is a circle with midpoint $D$ ...
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2answers
75 views

Compute coordinates of a point in 3D-Euclidean Space

My question concerns the computation of a point’s coordinates in three-dimensional Euclidean Space. I have a point P in three-dimensional Euclidean Space whose coordinates are unknown. My goal ...
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1answer
27 views

Triangles formed by diagonals of trapezoids

$\Delta$ AOB and $\Delta$ DOC should be equal in area. Correct me if I am wrong. Given: Trapezoid ABCD with ratio $\frac{area \Delta AOB}{area\Delta ABD}$ = $\frac{3}{4}$. I am trying to find (1) ...
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3answers
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Manhattan distance vs Euclidean distance

Suppose that for two vectors A and B, we know that their Euclidean distance is less than d. What can I say about their Manhattan distance?
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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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1answer
36 views

Simple geometric proof of parallel lines cut by transversals

Three parallel lines a,b and c are cut by transversal ABC. I need to prove that, if $AB = BC$, then $A'B' = B'C'$. I've made this drawing in geogebra. Any idea of what theorem is this? Could you ...
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1answer
29 views

Prove congruent angles have congruent supplements.

Prove congruent angles have congruent supplements. I do not yet have degrees. Could I somehow use the base angles of isosceles triangles are congruent?
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1answer
24 views

Prove vertical angles are congruent.

Prove vertical angles are congruent. I don't yet know degrees. All I know is congruent angles have congruent supplements. Is it too easy to just say that if I have two intersecting lines AC and BD ...
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4answers
325 views

Concentric Equilateral Triangles

I'm currently researching a particular dynamical system that is very geometric in nature. As part of this, I need to prove the following results (the second obviously implies the first). They are ...
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1answer
38 views

Find the inverse of defined operation $\Delta$

We defined the operation $ \Delta $ as $ (a,b) \Delta (c,d) = (ac + \delta bd, ad + bc) $ where $ a, b, c ,d \in \mathbb{Q} $ I have already proven that this operation is both commutative and ...
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2answers
40 views

Can you construct (ruler and compass) a square with an irrational area?

I've heard that when $\pi$ was proved irrational, that squaring the circle was not proved impossible. This lead me to believe that you could construct a square with an irrational area. Is this ...
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1answer
48 views

Prove every segment has a midpoint

Prove every segment has a midpoint. Unfortunately I do not have the definition yet of isosceles triangles. All I have is SSS and SAS. I also do not have right angles. But I do have perpendicular ...
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1answer
28 views

Prove every angle has a bisector.

Prove every angle has a bisector. I have successfully constructed a bisector and justified by construction. Now I need to put it in proof form. However, I technically do not know midpoints and ...
2
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1answer
36 views

How to show parallelism

Problem: Given two non-congruent circles that intersect at two points X and Y. One secant segment passes through X and intersects one circle (C1) at A and the other circle (C2) at B. Another secant ...
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1answer
59 views

Why must there be an infinite number of lines in an absolute geometry?

Why must there be an infinite number of lines in an absolute geometry? I see that there must be an infinite number of points pretty trivially due to the protractor postulate, and there are an infinite ...
3
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0answers
55 views

Pythagorean rectilinear polygons

Polygons all of whose edges meet at right angles are called rectilinear polygons. I am interested in rectilinear polygons with integer distance between each pair of vertices. Such rectilinear polygons ...
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1answer
29 views

Is it possible to determine triangle with prescribed centres (incentre, orthocentre, barycentre etc.)

The centres of a triangle is related to the triangle itself, or in the language of coordinate geometry, their coordinates can be calculated from that of the triangle's vertices. Can we reverse this ...
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invariant points of an isometry

If $A$ and $B$ are two invariant points of an isometry $f$, then every points in line $AB$ is an invariant point of $f$.
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103 views

Is Euclidean Geometry studied at all?

Is there a place for Euclidean geometry in the hearts or minds of any mathematicians? I personally find it to be the most beautiful mathematics I have yet encountered but I see little of it on sites ...
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1answer
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How one can show Gerretsen's inequality?

I read from http://rgmia.org/papers/v6n3/wsh.pdf the following: A triangle with semiperimeter $s$, circumradius $R$ and inradius $r$ satisfies $$16Rr-5r^2\leq s^2\leq 4R^2+4Rr+3r^2.$$ How can I prove ...
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1answer
16 views

Difference between $\mathbb{R}^2$ and $SE(2)$

I would like to have a good explanation of which is the difference between the Euclidean Group $SE(2)$ and the Euclidean space $\mathbb{R}^2$. From what I understood in $SE(2)$ there is also a ...
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1answer
29 views

How can we minimize this distance?

Given $A = (p, q)$ and $C = (−q, p)$ a pair of points in $\mathbb{R}^2$. Assume that $q > p > 0$. Find $x, y ∈ R$ such that for $ B = (x, 0), D = (0, y)$, $S = AB + BC − |CD − DA|$ is the ...
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4answers
173 views
+50

Is $540^\circ$ a straight angle?

The usual definition of a straight angle is a $180^\circ$ angle. however, because a $540^\circ$ angle is also the same shape, is it a straight angle as well?