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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22
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2answers
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Does there exist a copy of Euclid's Elements with modern notation and no figures?

I am working through Euclid's Elements for fun, but I find the propositions difficult to understand without referencing the provided figures. Unfortunately, the figures usually give away the proofs, ...
3
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4answers
1k views

Area Between Three Circles of Differing Radii

From the link in wikipedia http://web.gnowledge.org/wiki/index.php/Area_Between_Three_Circles_of_Differing_Radii OPEN QUESTION: What is the equation, in three variables, relating the radii of ...
17
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6answers
839 views

Geometrical construction for Snell's law?

Snell's law from geometrical optics states that the ratio of the angles of incidence $\theta_1$ and of the angle of refraction $\theta_2$ as shown in figure1, is the same as the opposite ratio of the ...
0
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1answer
32 views

ABC is a triangle, D is a point in the triangle. E is the midpoint of BD. AB=BC, angle ABD= angle DBC=35 degrees, angle ACD=25 degrees. Angle BAE=?

I tried to solve this problem but couldn't. I just know that here, angle BDC= 100 degrees, angle BAC= 40 degrees, AB^2+AD^2=2(AE^2+BE^2) and AB/AD={sin(angle DAE)}/{sin(angle BAE)}
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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 ...
0
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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 ...
3
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3answers
78 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?
3
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3answers
68 views

5 points on a plane with rational distances

Can you find 5 points on a plane whose Euclidean distances between them are all rational numbers and no 3 points out of them are co-linear? If the answer is yes, can we find a construction for ...
1
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2answers
70 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 ...
0
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1answer
38 views

Given $4$ points in the space, how do you check if an arbitrary point is within the area marked by those points?

Given $4$ arbitrary points in the space $A(x_1,y_1), B(x_2,y_2), C(x_3,y_3,), D(x_4,y_4)$, how do you check if an arbitrary point $X(x_5,y_5)$, is within the quadratic area marked by the $4$ points ...
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2answers
52 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 $D$ ...
8
votes
4answers
347 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 ...
1
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2answers
124 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 ...
1
vote
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) ...
1
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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 ...
0
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1answer
45 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 ...
5
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1answer
1k views

IMO 2014 problem 3, first day

Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such ...
0
votes
1answer
33 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?
0
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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 ...
0
votes
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 ...
0
votes
2answers
48 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 ...
2
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1answer
55 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 ...
0
votes
1answer
43 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
votes
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 ...
1
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1answer
64 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 ...
1
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2answers
9k 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 ...
3
votes
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
votes
1answer
32 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 ...
4
votes
2answers
142 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 ...
2
votes
1answer
287 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 ...
1
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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 ...
0
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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 ...
3
votes
2answers
108 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 ...
3
votes
3answers
267 views

Alternative model of Euclidean geometry

I'm planning to teach high-school geometry. As usual, this will be by building from axioms. (The axioms used are AFAICT particular to the book I've been assigned, but they're some combination of ...
0
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1answer
32 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 $l_2$ ...
3
votes
2answers
92 views

Prove that OD is a the angle bisector of the angle BOC.

Let $ABC$ be a non-isosceles triangle and $I$ be the intersection of the three internal angle bisectors. Let $D$ be a point of $BC$ such that $ID \perp BC$ and $O$ be a point on $AD$ such that $IO ...
1
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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 ...
2
votes
1answer
47 views

Name of theorem about two quadrilaterals with parallel edges

I'm looking for a name for the following theorem: If $abAB$ lie on one line and $cdCD$ lie on another line, and furthermore $ac\Vert AC,ad\Vert AD,bc\Vert BC$, then $bd\Vert BD$. One can ...
3
votes
0answers
229 views

Ellipse and circles

Playing with an ellipse I discovered the following properties and am now looking for a nice proof or references. Let's consider an ellipse with foci $A$ and $B$ and let $C$ be a point on it. Let $l$ ...
0
votes
1answer
55 views

Triangle $ABC$ and equilateral triangles $ABC'$, $BCA'$ and $ACB'$.

We consider a triangle $ABC$ whose angles are less then $120°$ and construct the equilateral triangles $ABC'$, $BCA'$ and $ACB'$, exterior to $ABC$. $I$ denotes the intersection of $(AA')$ and ...
0
votes
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 ...
1
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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?
0
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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 ...
4
votes
3answers
1k views

How to compute the volume of intersection between two hyperspheres

Let's say I have two n-spheres and I've no prior knowledge about the spheres (such as one of the sphere might be inside the other one) and I need to compute the volume of the intersection of the two ...
1
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1answer
46 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 ...
0
votes
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 ...
0
votes
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'. ...
14
votes
6answers
747 views

Why do we use the Euclidean metric on $\mathbb{R}^2$?

On the train home, I thought I would try to prove $\pi$ is irrational. I needed a definition, so I used: $\pi$ is the area of the unit circle. But what is a circle? A circle is the set of tuples ...
1
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1answer
45 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 ...
0
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1answer
25 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 ...