The combinatorial structure of trigonometry.
The combinatorially regular polyhedra of index 2.
The computation of certain numbers using a ruler and compass.
The edge-minimal polyhedral maps of Euler characteristic .
The envelope of the Wallace-Simpson lines of a triangle. A simple proof of the Steiner problem on the deltoid.
A simple proof is presented of a famous, and difficult, theorem by Jakob Steiner. By means of a straightforward transformation of the triangle, the proof of the theorem is reduced to the case of the equilateral triangle. Several relations of the Steiner deltoid with the Feuerbach circle and the Morley triangle appear then as obvious.
The Euler Line Revisited.
The generalized sine law and some inequalities for simplices.
The Gromov Invariant of Links.
The Hahn-Banach theorem implies the Banach-Tarski paradox
The hyperbolic Carnot theorem in the Poincaré disc model of hyperbolic geometry.
The hyperbolic triangle centroid
Some gyrocommutative gyrogroups, also known as Bruck loops or K-loops, admit scalar multiplication, turning themselves into gyrovector spaces. The latter, in turn, form the setting for hyperbolic geometry just as vector spaces form the setting for Euclidean geometry. In classical mechanics the centroid of a triangle in velocity space is the velocity of the center of momentum of three massive objects with equal masses located at the triangle vertices. Employing gyrovector space techniques we find...
The inner-product of iso-taxicab geometry.
The Lehmus inequality.
The length of a lemmiscate.
The Möbius-Pompeiu metric property.
The n -Point Condition and Rough CAT(0)
We show that for n ≥ 5, a length space (X; d) satisfies a rough n-point condition if and only if it is rough CAT(0). As a consequence, we show that the class of rough CAT(0) spaces is closed under reasonably general limit processes such as pointed and unpointed Gromov-Hausdorff limits and ultralimits.
The Neuberg cubic in locus problems.
The nine points circle and Euler's line. (El círculo de los nueve puntos y la recta de Euler.)
The number of triangles formed by intersecting diagonals of a regular polygon.
The optimal ball and horoball packings to the Coxeter honeycombs in the hyperbolic -space.