A porism for cyclic quadrilaterals, butterfly theorems, and hyperbolic geometry

If there exists a cyclic quadrilateral whose sides go through the given four collinear points, then there are infinitely many such quadrilaterals inscribed in the same circle. We give two proofs of this porism; one based on cross-ratios, the other on compositions of hyperbolic isometries.Comment: 9 pages, 10 figures, to appear in Amer. Math. Monthl

Integrative concept of homeostasis: translating physiology into medicine

To truly understand living systems they must be viewed as a whole. In order to achieve this and to come to some law to which living systems obey, data obtained on cells, tissues and organs should be integrated. Because there are no such laws yet, there is usually a long path for physiological findings obtained by reductionist approaches to be translated into medical practice. The concept and accompanying equations of homeostasis presented here are aimed to develop biological laws and to bridge this gap between physiology and medicine. The concept of homeostasis takes into account energy input and output, enlisting all relevant contributors. In homeostasis, input should equal the output. What I suggest here is that if the system is out of homeostasis, the homeostasis may be regained by changing any of the input or output components in an adequate manner, not only the one that has changed first. The proposed equation should enable for new lab findings regarding any pathophysiological conditions to find a more direct use in medicine. It should also ease ‘decision making’ in medicine and make therapy development and treatment outcome more straightforward and predictable. Finally, to recognize the basic laws of living systems enables for evolutionary adaptations and processes to be understood better