# Questions tagged [euclidean-geometry]

Geometry assuming the parallel postulate: 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.

625 questions
Filter by
Sorted by
Tagged with
56k 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 ...
58k 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 ...
12k 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)?
19k 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 ...
4k 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 ...
1k views

### A conjecture related to a circle intrinsically bound to any triangle

Given a triangle $ABC$, whose (one of the) longest side is $AC$, consider the two circles with centers in $A$ and $C$ passing by $B$. (The part in italic is edited after clever observations pointed ...
547 views

### Looking for an alternative proof of the angle difference expansion

I have thought about this for a while and have no progress. Does there exist a purely Euclidean Geometric proof of the Angle Difference expansion for Sine and Cosine, for Obtuse angles?
206 views

### Show that the angles satisfy $x+y=z$

How can I show that $x+y=z$ in the figure without using trigonometry? I have tried to solve it with analytic geometry, but it doesn't work out for me.
19k 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 ...
2k views

1k views

### If $J$ is tangent point of $GH$ with incircle of $FGH$ and $D$ is intersection of $F$-mixtilinear inclrcle with $(FGH)$, then $\angle FGH=\angle GDJ$.

Let $FGH$ be a triangle with circumcircle $A$ and incircle $B$, the latter with touchpoint $J$ in side $GH$. Let $C$ be a circle tangent to sides $FG$ and $FH$ and to $A$, and let $D$ be the point ...
30k views

### Difference between Euclidean space and vector space?

I often hear them used interchangeably ... they are very complicated to make any use of. Wikipedia words: Euclidean space: One way to think of the Euclidean plane is as a set of points satisfying ...
3k 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 "believe"...
2k views

### A conjecture involving prime numbers and parallelograms

As already introduced in this post, given the series of prime numbers greater than $9$, let organize them in four rows, according to their last digit ($1,3,7$ or $9$). The column in which they are ...