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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43
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15answers
11k 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 ...
20
votes
7answers
3k 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)?
15
votes
7answers
8k 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 ...
9
votes
3answers
12k 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 ...
27
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 ...
11
votes
5answers
3k 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 ...
5
votes
3answers
3k 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 ...
2
votes
2answers
634 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 ...
16
votes
5answers
904 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 ...
15
votes
5answers
1k 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
4answers
616 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 ...
8
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 ...
12
votes
2answers
282 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 ...
12
votes
5answers
879 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
351 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 ...
7
votes
3answers
993 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 ...
3
votes
2answers
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 ...
7
votes
2answers
437 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 ...
2
votes
1answer
156 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 ...
90
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. ...
26
votes
3answers
738 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
64k 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?
13
votes
2answers
609 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 ...
11
votes
3answers
371 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 ...
7
votes
2answers
811 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 ...
2
votes
2answers
167 views

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

How in this situation (presented in image) can I prove that $|CA|+|CB|=2|AB|$?
13
votes
3answers
12k 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 ...
7
votes
2answers
4k 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?
2
votes
4answers
1k 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 ...
1
vote
1answer
228 views

Euclidean Circle Geometry Problem

Let $\Gamma_1$ and $\Gamma_2$ be two non overlapping circles with centers $O_1$ and $O_2$ respectively. From $O_1$, draw the two tangents to $\Gamma_2$ and let them intersect $\Gamma_1$ at points $A$ ...
1
vote
4answers
2k 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
-1
votes
2answers
533 views

In △ ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees?

In △ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees? I know already that angle A and angle D are congruent because ...
12
votes
1answer
574 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 ...
7
votes
1answer
165 views

Rotation of $\mathbb{R}^3$ by using quaternion

Express the rotation of $\mathbb{R}^3$ by $\frac{\pi}{3}$ about the $x=y=z$ axis by using quaternions and identifying $\mathbb{R}^3$ with $(i,j,k)$-space. Thoughts: From my point of view, every ...
5
votes
2answers
661 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, ...
4
votes
1answer
99 views

metric on the Euclidean Group

I am not an expert in this so I hope this doesn't sound so stupid: what is the common metric used when studying the Euclidean Group $\mathrm{E}(3)$. One could probably ask the same thing for ...
4
votes
4answers
790 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?
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
1answer
625 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
3answers
178 views

Sum of coefficients of an orthogonal matrix

Let $(a_{ij})_{1 \le i,j \le n}$ be a real orthogonal matrix. Show that $$\left| \sum_{1 \le i,j \le n} a_{ij}\right| \le n.$$ Naively applying the Cauchy-Schwarz inequality only gives ...
1
vote
2answers
53 views

Connecting square vertexes with minimal road

I have four cities in $A=(0,0),B=(1,0),C=(1,1),D=(0,1)$. I am asked to build the shortest motorway to connect the cities. How can I do that? I was thinking that first I need some compactness argument ...
1
vote
1answer
224 views

Reducing the maximum euclidean distance

This question comes from the HackerRank's "20/20 Hack February" contest which has now ended (problem link). There are N bikers present in a city (shaped as a grid) having M bikes. All the bikers ...
1
vote
2answers
105 views

If $ABCD$ is a cyclic quadrilateral, then $AC\cdot(AB\cdot BC+CD\cdot DA)=BD\cdot (DA\cdot AB+BC\cdot CD)$

If $ABCD$ is a cyclic quadrilateral, then $$ AC\cdot(AB\cdot BC+CD\cdot DA)=BD\cdot (DA\cdot AB+BC\cdot CD) $$ I tried using many approaches, but I could not find a proper solution. Can anyone please ...
1
vote
1answer
596 views

How to find the intersection of the area of multiple triangles

I have a couple of questions regarding finding the intersection of triangles. I have a system of 16 projectors that all have slightly different color gamuts. The color gamuts are represented by a ...
1
vote
2answers
171 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 ...
1
vote
1answer
208 views

Triangle and rational numbers

A triangle has rational side lengths and rational angles measured as degrees. Is such a triangle necessary equilateral?
0
votes
1answer
118 views

In a convex Quadrilateral ABCD, $\angle ABC = \angle BCD = 120^{\circ}$.Prove that: $ AC + BD \ge AB + BC + CD$

In a convex Quadrilateral ABCD, $\angle ABC = \angle BCD = 120^{\circ}$.Prove that: $$ AC + BD \ge AB + BC + CD$$ My attempt Tried to use cosine formula twice' i.e ($\triangle ABC$ and ...
0
votes
1answer
238 views

Intersection of Two Simplices

How to find vertices a the polytope-intersection of two simplices, if I know the vertices of these simplices. More precisely: Let $T_1$ and $T_2$ be two regular $n-1$ dimensional simplices with ...
0
votes
0answers
130 views

Tetrahedron parallel planes feet of altitudes

Could somebody help me prove that the plane KLM, where K, L, M are feet of altitudes of an orthocentric tetrahedron ABCS inscribed in a sphere is parallel to the base ABC of this pyramid? I would ...
-1
votes
1answer
429 views

Using the three reflection theorem

There is this sample question from my book that I dont know how to go about. please help out Use the three reflections theorem to show that the only transformations of the Euclidean plane are ...