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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59
votes
17answers
32k views

What is the most elegant proof of the Pythagorean theorem? [closed]

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 ...
21
votes
8answers
5k 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
6answers
6k 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 ...
10
votes
3answers
22k views

Compass-and-straightedge construction of the square root of a given line?

Given A straight line of arbitrary length The ability to construct a straight line in any direction from any starting point with the "unit length", or the length whose square root of its magnitude ...
23
votes
7answers
17k 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 ...
15
votes
4answers
2k 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 ...
5
votes
2answers
5k views

Euler angles and gimbal lock

Can someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading ...
30
votes
3answers
2k 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 ...
7
votes
3answers
2k views

Why is the inradius of any triangle at most half its circumradius?

Is there any geometrically simple reason why the inradius of a triangle should be at most half its circumradius? I end up wanting the fact for this answer. I know of two proofs of this fact. Proof ...
21
votes
3answers
22k views

Check if a point is within an ellipse

I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane. How do I determine if a point $(x,y)$ is within the area bounded by ...
19
votes
6answers
2k 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 ...
1
vote
1answer
95 views

Do any books or articles develop basic Euclidean geometry from the perspective of “inner product affine spaces”?

Definitions. By a vector space, I simply mean an $\mathbb{R}$-module. By an affine space, I mean a vector space $X$ (the "translation space") together with a set $P$ (of "points"), together with an ...
2
votes
2answers
739 views

Euclidean Geometry Intersection of Circles

Two circles intersect in the Cartesian Coordinate system at points $A$ and $B$. Point $A$ lies on the line $y=3$. Point $B$ lies on the line $y=12$. These two circles are also tangent to the x-axis at ...
-1
votes
2answers
88 views

Bisecting line segments in a tetrahedron. [closed]

Suppose that $OABC$ is a regular tetrahedron with base $ABC$. Suppose further that $T$ is the mid-edge of $AC$, $Q$ is the mid-edge of $OB$, $P$ is the mid-edge of $OA$, and $U$ is the mid-edge of ...
103
votes
3answers
14k views

Slice of pizza with no crust

The following question came up at a conference and a solution took a while to find. Puzzle. Find a way of cutting a pizza into finitely many congruent pieces such that at least one piece of pizza ...
19
votes
6answers
1k 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 ...
10
votes
2answers
2k 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 ...
14
votes
2answers
388 views

Decomposable Families of Shapes

There are two types of golden triangles in the world, as shown in the following picture: Here $\varphi = \dfrac{1+\sqrt{5}}{2}$ denotes the golden ratio. Each of these golden triangles can be ...
14
votes
6answers
881 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 ...
12
votes
5answers
1k 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? ...
3
votes
3answers
386 views

Prove that CX and CY are perpendicular

There is given convex quadrilateral ABCD. And internal bisectors of angle $\angle A$ and $\angle C$ intersect in point X. And internal bisectors of angle $\angle B$ and $\angle D$ intersect in point ...
10
votes
3answers
6k 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?
8
votes
2answers
563 views

The Dido problem with an arclength constraint

It is well known that the solution to the classical Dido problem is a semicircle, and that the solution to the classical isoperimetric problem is a circle. It's also reasonably obvious that the ...
6
votes
3answers
2k 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 ...
2
votes
1answer
247 views

Probabilities of Non-Regular Dice

Thinking about dice: for all the Platonic solids, it's very easy to figure out the odds of a particular face landing face-up in a roll of the die. If I have an arbitrary 6-sided solid, how do you ...
4
votes
3answers
1k views

Distance between a point and a line in space

I have two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ on a line, $L$, and another point $P_0(x_0, y_0, z_0)$. I want to find the distance between $P_0$ and $L$. Could someone help?
1
vote
2answers
257 views

Is $\mathbb{R}^2$ minus a countable number of points 'skew-Manhattan connected'

Let $A \subset \mathbb{R}^2$ be countable. Then it is not too hard to show that $\mathbb{R}^2 \setminus A$ is path-connected. However it is not always Manhattan connected since if $A = \mathbb{Q}^2 ...
99
votes
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. ...
27
votes
4answers
1k views

Two squares in a box.

According to Arthur Engel, "Problem Solving Strategies", this problem goes back to Erdős, but I cannot find the solution: Let $A$ and $B$ be two non-overlapping squares inside a unit square, of side ...
17
votes
1answer
109k views

Solving Triangles (finding missing sides/angles given 3 sides/angles)

What is a general procedure for "solving" a triangle—that is, for finding the unknown side lengths and angle measures given three side lengths and/or angle measures?
14
votes
2answers
1k 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 ...
7
votes
1answer
241 views

How can we draw a line?

We want to join by a line two distinct points $A$ and $B$. We have only a ruler of length $l>0$ and a pen. If $AB>l$ how can we do this?
16
votes
4answers
1k views

The position of a ladder leaning against a wall and touching a box under it

I was reading a newspaper and there was a little math riddle, I thought "how funny, that's gonna be easy, let's do it" and here am I... The problem goes as follow : in a barn, there is a 1 meter ...
7
votes
1answer
138 views

Prove that every triangle is the orthogonal projection of an equilateral one

Prove that every triangle is the orthogonal projection of some equilateral triangle. This problem appears in a book I'm working through in the chapter on transformations in space. There is a rather ...
10
votes
3answers
2k views

Formal Proof that area of a rectangle is $ab$

I tried to prove that the area of a rectangle is $ab$ given side lengths $a$ and $b$. The best I can do is the assume the area of a $1\times1$ square is $1$. Then not the number of $1\times1$ squares ...
15
votes
4answers
758 views

Why do we believe the equation $ax+by+c=0$ represents a line?

I'm going for quite a weird question here. As we know, the equation in Cartesian coordinates for a line in 2-dimensional Euclidean geometry is of the form $ax+by+c=0$. I'm wondering why do we ...
4
votes
2answers
163 views

The concurrence of angle bisector, median, and altitude in an acute triangle

$ABC$ is an acute triangle. The angle bisector $AD$, the median $BE$ and the altitude $CF$ are concurrent. Prove that angle $A$ is more than $45$ degrees. Here $D,E,F$ are points on $BC,CA,AB$ ...
4
votes
3answers
293 views

Smallest square containing a given triangle

Given a triangle $T$, how can I calculate the smallest square that contains $T$? Using GeoGebra, I implemented a heuristic that seems to work well in practice. The problem is, I have no proof that it ...
2
votes
1answer
148 views

Two conics from six Thebault circles of a triangle

[Significant edits, context, etc, provided by Blue, based on comments from the original author.] While exploring Thebault circles in GeoGebra (see my interactive sketch), I noticed some interesting ...
2
votes
2answers
178 views

Proving that $|CA|+|CB|=2|AB|$ in a general $ABC$ triangle

How in this situation (presented in image) can I prove that $|CA|+|CB|=2|AB|$?
10
votes
1answer
2k views

Construction of a regular pentagon

In Robert Dixon's Mathographics, a regular pentagon is constructed with straightedge and compass only. It is the pentagon $ABCDE$ pictured below. I am having trouble seeing why the central angles ...
6
votes
3answers
19k views

Proof of Angle in a Semi-Circle is 90 degrees

There is a well known theorem often stated as the angle in a semi-circle being 90 degrees. To be more accurate, any triangle with one of its sides being a diameter and all vertices on the circle has ...
4
votes
1answer
88 views

Two conjectures of four squares

I found a conjecture of four squares since two months ago. But I don't have a solution for the conjecture. This conjecture is nice result in Euclidean geometry. I hope that there is a solution: ...
1
vote
4answers
5k views

Prove that if a, b, x, y are integers with ax + by = gcd(a, b) then gcd(x,y)= 1

How would I go about proving this: Prove that if a, b, x, y are integers with ax + by = gcd(a, b) then gcd(x,y)= 1
10
votes
5answers
731 views

Eritrea's Theorem

According to this newspaper, an Eritrean high school student named Saied Mohammed Ali has discovered a new geometric theorem. Another source seems to say that it's the following: Say you have a ...
6
votes
2answers
950 views

Aren't asteroids contradicting Euler's rotation theorem?

I am totally confused about Euler's rotation theorem. Normally I would think that an asteroid could rotate around two axes simultaneously. But Euler's rotation theorem states that: In geometry, ...
3
votes
1answer
1k views

How to prove the midpoint of a chord is also the midpoint of the line segment defined by the points of intersection of other two chords with it?

Bernhard Elsner, alias MathOMan, posted this exercise in plane Geometry, Theorem about a circle, three chords and a midpoint on January 29th, 2010. "Let $\mathcal{C}$ be a circle, $A,B$ two distinct ...
2
votes
2answers
126 views

Relationship between the side lengths of a tetrahedron and an inscribed tetrahedron with vertices at the centroids

Suppose that $OABC$ is a regular tetrahedron with sides having centroids $\lbrace E,F,G,H\rbrace$ also forming a regular tetrahedron. What is the relationship between the side lengths of $OABC$ and ...
2
votes
1answer
912 views

Similar Triangle Theorem in the Incommensurable Case

The following is a geometry theorem whose proof is examinable in the Irish 'High School' Exam. Let $\Delta ABC$ be a triangle. If a line $L$ is parallel to $BC$ and cuts $[AB]$ in the ratio $s:t$, ...
2
votes
4answers
2k views

Proving two lines trisects a line

A question from my vector calculus assignment. Geometry, anything visual, is by far my weakest area. I've been literally staring at this question for hours in frustrations and I give up (and I do mean ...