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.
78
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4answers
1k 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.
...
58
votes
4answers
1k views
Probability that a stick randomly broken in five places can form a tetrahedron
Randomly break a stick in five places.
Question: What is the probability that the resulting six pieces can form a tetrahedron?
Clearly satisfying the triangle inequality on each face is a necessary ...
31
votes
11answers
4k views
What is the most elegant proof of the Pythagorean theorem?
The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield).
What's the most elegant proof?
My favorite ...
23
votes
1answer
486 views
About Euclid's Elements and modern video games
I just watched this video about Euclid's treatise the Elements. I got introduced to the postulates and a couple of propositions of book I. I really liked this video, I'm not sure if this is because of ...
22
votes
3answers
453 views
Two squares in a box.
According to Arthur Engel, "Problem Solving Strategies", this problem goes back to Erdös, but I cannot find the solution:
Let $A$ and $B$ be two non-overlapping squares inside a unit square, of side ...
21
votes
3answers
1k views
Why is Euclidean geometry scale-invariant?
In Euclidean geometry, I frequently use concepts related to invariance under scaling. For example, I know that if two squares have different side lengths, the ratio of their side lengths is the ...
17
votes
2answers
908 views
Intersection of two parabolae
Problem:
Consider two parabolae such that their axes of symmetry form a right angle. Prove that all four points of intersection lie on a common circle (it is an assumption that there exist such four ...
16
votes
3answers
425 views
Projection of tetrahedron to complex plane
It is widely known that:
distinct points $a,b,c$ in the complex plane form equilateral triangle iff $ (a+b+c)^{2}=3(a^{2}+b^{2}+c^{2}). $
New to me is this fact:
let $a,b,c,d$ be the images of ...
15
votes
1answer
243 views
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, ...
14
votes
1answer
793 views
Can a regular heptagon be constructed using a compass, straightedge, and angle trisector?
Euclid has a magical compass with which he can trisect any angle. Together with a regular compass and a straightedge, can he construct a regular heptagon?
14
votes
1answer
316 views
Two tetrahedra are congruent given a certain condition
This question is inspired by a Miklos Schweitzer problem, namely
Problem 9./2007 Let $A$ and $B$ be two triangles in the plane such that the interior of both triangles contains the origin, and for ...
13
votes
3answers
333 views
If any triangle has area at most 1 , points can be covered by a rectangle of area 2.
I am working on this problem for some time, and I am not able to finish the argument:
There is a finite number of points in the plane, such that every triangle has area at most 1. Prove that the ...
13
votes
3answers
295 views
What is the expected area of a polygon whose vertices lie on a circle?
I came across a nice problem that I would like to share.
Problem: What is expected value of the area of an $n$-gon whose vertices lie on a circle of radius $r$? The vertices are uniformly ...
13
votes
3answers
274 views
How many 1-balls are needed to cover the 2-ball in an $n$-dimensional Euclidean Space?
Consider $\mathbb{R}^n$ and its usual Euclidean norm given by the distance $d(x,y) = \sqrt{\sum_{i=1}^n (x_i-y_i)^2}$.
Let $B(y,1) = \{ x\in \mathbb{R}^n : d(x,y) \leq 1 \}$ be the closed 1-ball ...
12
votes
6answers
2k views
How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$?
I would like to find the apothem of a regular pentagon. It follows from
$$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$
But how can this be proved (geometrically or trigonometrically)?
12
votes
5answers
946 views
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 ...
12
votes
2answers
504 views
Finding the circles passing through two points and touching a circle
Given two points and a circle, construct a/the circle through the two points and
touching the given circle.
I came across this problem in History of Numerical Analysis by H. Goldstein. I
spent some ...
12
votes
1answer
318 views
What's the average width of a convex polygon?
If one computes the average width of a triangle, then one gets $(s_1+s_2+s_3)/\pi$, where $s_1$, $s_2$, $s_3$ are the side lengths. I did this by brute force, using an integral which went through an ...
12
votes
2answers
183 views
6 point lying on a common circle
$Z$ is an interior point of segment $XY$. Three semicircles are drawn over segments $XY$, $XZ$ and $ZY$ on the same side. The midpoints of the arcs are $M1$, $M2$ and $M3$ respectively. A circle ...
11
votes
5answers
672 views
Why the interest in locally Euclidean spaces?
A lot of mathematics as far as I know is interested in the study of Euclidean and locally Euclidean spaces (manifolds).
What is the special feature of Euclidean spaces that makes them interesting?
...
11
votes
1answer
242 views
Sub-determinants of an orthogonal matrix
Let $A$ be a matrix in the special orthogonal group, $A \in SO_n$. This means that $A$ is real, $n \times n$, $A^t A = I$ and $Det(A)=1$, that is, the column vectors of $A$ make a positively-oriented ...
11
votes
1answer
286 views
What regular polygons can be constructed on the points of a regular orthogonal grid?
Besides a square, what regular polygons can be constructed so that the points of that polygon lie on the points of a regular, planar, orthogonal grid?
Besides a triangle and hexagon, what regular ...
10
votes
2answers
388 views
If the Minkowski sum of two convex closed sets is a Euclidean ball, then can the two sets be anything other than Euclidean balls?
If for two convex closed sets $S_1$ and $S_2$, the Minkowski sum is a Euclidean ball then can $S_1$ and $S_2$ be anything other than Euclidean balls themselves. I suspect they can be but I haven't ...
10
votes
6answers
1k views
Where does the Pythagorean theorem “fit” within modern mathematics?
I am interested in how today's professional mathematicians view the Pythagorean theorem, in terms of how the theorem fits within the axiomatic framework of mathematics. I often come across textbooks ...
10
votes
3answers
404 views
What is a point?
In geometry, what is a point?
I have seen Euclid's definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know ...
10
votes
2answers
584 views
Software for solving geometry questions
When I used to compete in Olympiad Competitions back in high school, a decent number of the easier geometry questions were solvable by what we called a geometry bash. Basically, you'd label every ...
10
votes
1answer
280 views
A hard problem from regular $n$-gon
I found the following theorem in a book of mine without a proof. Could someone show me a proof of it?
Given a regular $n$-gon, with $n$ odd and vertices $v_1,\ldots,v_n$, and $C$ its circumcircle. At ...
10
votes
2answers
340 views
Decomposing the plane into intervals
A recent Missouri State problem stated that it is easy to decompose the plane into half-open intervals and asked us to do so with intervals pointing in every direction. That got me trying to ...
10
votes
1answer
292 views
Proving collinear points
This problem is so hard that I cannot figure it out. I hope you guys can give me a small push on how to tackle this problem, as I have been thinking about this for, like a week. Here's the problem:
...
9
votes
5answers
439 views
What are the postulates that can be used to derive geometry?
What are the various sets of postulates that can used to derive Euclidean geometry?
It might be nice to have several different approaches together for comparison purposes and for ready reference.
It ...
9
votes
4answers
266 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 ...
9
votes
1answer
172 views
A curve that intersects every plane in finitely but arbitrarily many points
Does there exist a piecewise smooth curve in $\mathbb{R}^3$ such that every plane intersects the curve at finitely many points and the number of intersection points can be arbitrary large?
If the ...
9
votes
2answers
387 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 ...
9
votes
2answers
930 views
Proof that every polygon with an inscribed circle is convex?
In many elementary (and not-so-elementary) Euclidean geometry texts, a (simple) polygon is said to be tangential if it is convex and has an inscribed circle (i.e., a circle that intersects and ...
8
votes
1answer
304 views
Is there a value for $\pi$ that relates to triangles?
So I heard that if one inscribes the largest circle that can fit into a equilateral triangle, then divides the perimeter of the triangle by the diameter of the inscribed circle, it gives a value which ...
8
votes
5answers
401 views
What is the modern axiomatization of (Euclidean) plane geometry?
I have heard anecdotally that Euclid's Elements was an unsatisfactory development of geometry, because it was not rigorous, and that this spurred other people (including Hilbert) to create their own ...
8
votes
3answers
734 views
Minimal Ellipse Circumscribing A Right Triangle
Find the equation of the ellipse circumscribing a right triangle whose lengths of it's sides are $3,4,5$ and such that its area is the minimum possible one.
You may chose the origin and orientation ...
8
votes
5answers
827 views
Geometry Book Recommendation?
Can someone recommend a good basic book on Geometry? Let me be more specific on what I am looking for. I'd like a book that starts with Euclid's definitions and postulates and goes on from there to ...
8
votes
3answers
540 views
Finding angles in a parallelogram without trigonometry
I'm wondering whether it's possible to solve for $x^{\circ}$ in terms of $a^{\circ}$ and $b^{\circ}$ given that $ABCD$ is a parallelogram. In particular, I'm wondering if it's possible to solve it ...
8
votes
2answers
445 views
Sangaku: Show line segment is perpendicular to diameter of container circle
"From a 1803 Sangaku found in Gumma Prefecture. The base of an isosceles triangle sits on a diameter of the large circle. This diameter also bisects the circle on the left, which is inscribed so that ...
8
votes
2answers
400 views
geometric meaning of a trigonometric identity
It follows from the law of cosines that if $a,b,c$ are the lengths of the sides of a triangle with respective opposite angles $\alpha,\beta,\gamma$, then
$$
a^2+b^2+c^2 = 2ab\cos\gamma + 2ac\cos\beta ...
8
votes
1answer
222 views
Chebyshev center = center of mass?
I would like to know for which convex polyhedra $P$ in $\mathbb{R}^3$,
is the center of the largest sphere enclosed in $P$ (a.k.a. the Chebyshev center, or the incenter)
the same as the center of ...
8
votes
0answers
122 views
Who first discovered that the torus supports a flat structure?
Who first recognized that there exists a homogenous metric on the closed genus 1 orientable surface?
8
votes
0answers
162 views
Can we alter Hilbert's axioms to have $\mathbb{Q}^3$ as a unique model?
Can we alter Hilbert's axioms to have $\mathbb{Q}^3$ as a unique model?
The critical axioms seem to be the congruence axioms IV.1 and IV.4, and presumably the line completeness axiom V.2.
But how ...
7
votes
2answers
1k views
What is wrong in my proof that 90 = 95? Or is it correct?
Hi I have just found the proof that 90 equals 95 and was wondering if I have made some mistake. If so, which step in my proof is not true?
Definitions:
1. $\angle ABC=90^{\circ}$
2. $\angle ...
7
votes
6answers
2k views
Book recommendation on plane Euclidean geometry
I consider myself relatively good at math, though I don't know it at a high level (yet). One of my problems is that I'm not very comfortable with geometry, unlike algebra, or to restate, I'm much more ...
7
votes
2answers
674 views
Why Euclidean geometry cannot be proved incomplete by Gödel's incompleteness theorems?
Wikipedia mentioned the limitation of Gödel's theorems. According to it, Euclidean geometry doesn't satisfy the hypotheses of Gödel's incompleteness theorems, that is, it cannot define natural ...
7
votes
3answers
526 views
Find the perimeter of any triangle given the three altitude lengths
The altitude lengths are 12, 15 and 20. I would like a process rather than just a single solution.
7
votes
6answers
1k views
What is the Direction of a Zero (Null) Vector?
To be more precise, I am interested in knowing if the intuition that a Euclidean zero vector does not have a particular direction is actually correct, and if there is a rigorous formulation that would ...
7
votes
2answers
991 views
The shortest distance between any two distinct points is the line segment joining them.How can I see why this is true?
On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true?
