The Droz-Farny Circles of a Convex Quadrilateral

On a Porism Associated with the Euler and Droz-Farny Lines

The envelope of the Droz-Farny lines of a triangle is determined to be the inconic with foci at the circumcenter and orthocenter by using purely Euclidean means. The poristic triangles sharing this inconic and circumcircle have a common circumcenter, centroid and orthocenter.

A Purely Synthetic Proof of the Droz-Farny Line Theorem

Angle and Circle Characterizations of Tangential Quadrilaterals

We prove five necessary and sufficient conditions for a convex quadrilateral to have an incircle that concerns angles or circles.

A convex combinatorial property of compact sets in the plane and its roots in lattice theory

K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ ...

Diamond-Kite Meshes: Adaptive Quadrilateral Meshing and Orthogonal Circle Packing

We describe a family of quadrilateral meshes based on diamonds, rhombi with 60◦ and 120◦ angles, and kites with 60◦, 90◦, and 120◦ angles, that can be adapted to a local size function by local subdivision operations. The vertices of our meshes form the centers of the circles in a pair of dual circle packings in which each tangency between two circles is crossed orthogonally by a tangency betwee...