miércoles, 17 de mayo de 2023

A Generalization of the Law of Sines

 The following is an extension of the law of sines for cyclic quadrilaterals that comes to accompany the generalization of Mollweide's formula (rather Newton's) and the generalization of the law of tangent.

Generalization. Consider a cyclic quadrilateral, $ABCD$, with side lengths $AB=a$, $BC=b$, $CD=c$, and $DA=d$. Let $\angle{DAB}=\alpha$, $\angle{ABD}=\beta$, $\angle{BCD}=\gamma$, and $\angle{CDA}=\delta$. Then the following identity holds


At first, I was reluctant to publish this result because the proof is very straightforward (hence, I leave it as an exercise to the reader). However, I have shared it on my social media platforms (see herehere and here) and the audience's response has been more favorable than my generalizations of Mollweide's formula and the law of tangents combined.

No hay comentarios:

Publicar un comentario