1
vote
1answer
99 views

Angles and projective metric

Unless I am very wrong, the following seems to be true: If the angle between two vectors in $\mathbb{R}^{n}_{++}$ is small, then the value of the Hilbert projective metric between them is also ...
0
votes
1answer
149 views

A little question on certain parallel-lines-preserving maps

Let $\alpha:\mathbb{R}^n\to\mathbb{R}^n$, $n\geq 2$, be a $\mathbb{Q}$-linear bijection with the following properties: 1) $\alpha$ sends straight affine $\mathbb{R}$-lines to straight affine ...
18
votes
6answers
1k views

Does every ellipse inside a tetrahedron inside a ball fit in a triangle inside the ball?

In three-dimensional euclidean space, consider the closed unit ball $B$. Let $T$ be a tetrahedron, and $E$ an ellipse, with $E \subset T \subset B$. Does there necessarily exist a triangle $T'$ with ...
2
votes
3answers
1k views

How to solve geometry problems using involutions

Some geometry problems ( like this and this ) have short solutions if we use involutions. What references are there for solving geometry problems using involutions? I am particularly interested in ...
14
votes
2answers
914 views

Why do all incidence theorems follow from Pappus' theorem?

In Hilbert and Cohn-Vossen's ``Geometry and the Imagination," they state in the last paragraph of Chapter 20 that "Any theorems concerned solely with incidence relations in the [Euclidean ...