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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34 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
69 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
110 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
15 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
38 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
47 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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0answers
36 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
43 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
29 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
40 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
29 views

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
41 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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2answers
104 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
71 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
132 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
55 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
127 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
43 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
80 views

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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0answers
16 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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1answer
47 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
35 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
28 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
349 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
42 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
50 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
57 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
45 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
38 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
65 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 ...
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0answers
56 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 ...
2
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1answer
36 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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2answers
143 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
64 views

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
19 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
30 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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6answers
253 views

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?
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1answer
33 views

How should we minimize the area of this triangle?

Consider the angle between two rays $l_1$ and $l_2$ with a vertex O and point A in this angle. Now Consider all possible triangles with vertex O such that two sides of them belong to $l_1$ and $l_2$ ...
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1answer
35 views

Is it true that these angles are equal?

Suppose we have a line $l$ and points $A$ and $B$ which are on different sides of $l$. Point $P$ is on line $l$. When we maximize $|PA-PB|$, it seems that the angle formed by $PA$ an $l$ is equal to ...
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1answer
19 views

Value of X for which both tha angles are equal?

${C}$ and ${D}$ are two points on the same side of a straight line ${AB}$. Find a point X on AB such that the angles ${CXA}$ and ${DXB}$ are equal. Note: This is how I have approached the problem. We ...
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1answer
41 views

Perimeter of triangle formed by connecting intersection points of altitudes.

Given acute triangle $ABC$ with altitudes $AA_1, BB_1,CC_1$. How do we show that the perimeter of triangle $A_1B_1C_1$ is less than twice the length of any of the altitudes?
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2answers
110 views

How do I deal with reflections inside an ellipse?

Suppose I have an ellipse with foci $F_1$ and $F_2$. How do I show that any ray of light which intersects the segment connecting the foci will have subsequent reflections that always are tangent to ...
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1answer
38 views

A level Ellipse question

An ellipse has the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where $a>b$, and with eccentricity $e$. It also has foci $S$ and $S'$ and directrices $l$ and $l'$. a) Use the focus-directrix ...
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1answer
47 views

How do I find the area of a triangle formed by cevians?

Given $\triangle ABC$, locate points $A_1$, $B_1$, $C_1$ on respective sides $BC$, $CA$, $AB$ such that $$\frac{BA_1}{A_1C} =\frac{CB_1}{B_1A} = \frac{AC_1}{C_1B} = 2$$ How can I show that the ...
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1answer
38 views

Prove that for any two points $A$ and $B$ $\overrightarrow{AB} \cup \overrightarrow{BA} = \overleftrightarrow{[AB]}$

Question: Prove that for any two points $A$ and $B$ $\overrightarrow{AB} \cup \overrightarrow{BA} = \overleftrightarrow{[AB]}$ The right hand side of the statement that I am trying to prove is a line ...
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1answer
31 views

What can we say about the areas of these two triangles?

Given triangle ABC. Let X be a point on AB, Y be a point on BC and Z be a point on AC. Now suppose we reflect X, Y, Z around the midpoint of the sides they are on and label the images X', Y' and Z'. ...
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1answer
46 views

Projective Geometry with Clifford Algebra - lost Inner Product

Projective geometry may be studied with the tools of Clifford Algebras by adding a new direction (see for example this article). But as far as I understand it, only blades and null spaces are used for ...
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1answer
26 views

Models of Incidence Geometry

First I'd like to thank you for reading this, there's a lot here. Secondly I am having a lot of trouble negating the first axiom which I think is hindering my ability to apply it to these types of ...
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3answers
58 views

Prove that if $l$ is a line in the classical euclidean plane, then there is a point $p$ that lies on $l$

Suppose that $\mathbb{P}$ is a Classical Euclidean Plane (satisfies all five of Euclid's postulates). Can you prove that if $L$ is a line in $\mathbb{P}$, then there is at least one point $p$ in ...