# Questions tagged [euclidean-geometry]

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these.

444 questions
Filter by
Sorted by
Tagged with
1 vote
86 views

386 views

### Maximizing the area of a region involving triangles

I thought of a question while making up an exercise sheet for high school students, and posted it on MathStackExchange but did not receive an answer (the original post is here), so I thought perhaps ...
267 views

### A randomized version of straight-edge and compass construction

Suppose you start with two points in the plane which are distance 1 apart, which for concreteness can be $(0,0)$ and $(0,1)$. Then you keep marking new points based on ruler and compass constructions. ...
1 vote
71 views

### Pseudo-Droz-Farny circles

I would like to present a construction of 2 circles. These 2 circles are somewhat similar in appearance to the well known Droz-Farny circles that can be drawn for every isogonal conjugate pairs of ...
1 vote
72 views

### Definition of a unit ball in an Euclidean subspace? [closed]

Suppose $\Lambda$ is a $3$ dimensional lattice inside $\mathbb{R}^4$ and let $E$ be the subspace $\mathbb{R}$-spanned by $\Lambda$. What exactly is meant by the unit ball in $E$? This is something ...
221 views

I made some rookie attempt to define the 4th Lemoine circle recently. The alternative name for this circle was suggested yesterday. Further investigation revealed a family of circles associated with ...
1 vote
103 views

### How do Euclid’s postulates imply that a line has more than two points? [closed]

How do Euclid’s postulates imply that there exist more points than provided as assumptions, e.g. in the statement: Let C1 be the circle centered at A, with radius AB, let C2 be the circle centered at ...
53 views

### Historical question about tangent lines to disjoint circles

It is pretty well known that two disjoint circles have 4 different lines that are simultaneously tangent to both circles. There are constructions with ruler and compass available in many books, but I ...
7k views

### The 4th vertex of a triangle?

I was immensely surprised and amused by the idea of the fourth side of a triangle that was introduced by B.F.Sherman in 1993. 'Sherman's Fourth Side of a Triangle' by Paul Yiu is available here. ...
213 views

### Constructing an equilateral triangle using Tarski's axioms for geometry

In Euclid's first geometry proposition, he constructs an equilateral triangle given an arbitrary line segment. I was wondering if it was possible to prove this straight from Tarski's axioms for ...
345 views

### The 4th Lemoine circle

The first and second Lemoine circles are well-known to geometers. According to this article the third Lemoine circle has been first discovered by Jean-Pierre Ehrmann in 2002. It is worth noting that ...
842 views

### Is there a square with all corner points on the spiral $r=k\theta$, $0 \leq \theta \leq \infty$?

I've posted this question on Math Stack Exchange, but I want to bring it here too, because 1) the proof seems missing in the literature, although they are some sporadic mentions and 2) maybe it ...
1 vote
63 views

### Collinearity in tangential pentagon [closed]

I am looking for a proof of the following claim: Given tangential pentagon. Touching point of the incircle and the side of the pentagon,the vertex opposite to that side and the intersection point of ...
418 views

### Is there a mapping from Euclidean space to real numbers which relatively preserves distance? [closed]

Motivation: I need to find a mapping from $n$-dimensional Euclidean space to real numbers such that the distance between each pair of points in the quoted space is relatively-preserved after the ...
129 views

### The product of the lengths of two line segments that belong to Newton line [closed]

I am looking for the proof of the following claim: Consider a family of bicentric quadrilaterals with the same inradius length and the same distance between incenter and circumcenter. Denote by $P$ ...
402 views

### Is this a valid definition of Euclidean geometry? [closed]

In trying to understand what actually constitutes a "geometry" I came across many definitions of Euclidean spaces and geometries. Euclidean space is defined as an affine space with an inner ...