It is well known that Heron’s equality provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its edges. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges. Surprisingly, this has proved to be far from simple, and no explicit solutions exist for cyclic polygons having \(n>4\) edges. In this paper we investigate such a problem by following a new and elementary approach, based on the idea that the simple geometry underlying Heron’s and Brahmagupta’s equalities hides the real players of the game. In details, we propose to focus on the dissection of the edges determined by the incircles of a suitable triangulation of the cyclic polygon, showing that this approach leads to an explicit formula for the area as a symmetric function of the lengths of these segments. We also show that such a symmetry can be rediscovered in Heron’s and Brahmagupta’s results, which consequently represent special cases of the provided general equality.

众所周知，Heron等式为三角形的面积提供了一个明确的公式，作为三角形边缘长度的对称函数。Brahmagupta已将其扩展为刻在一个圆上的四边形（循环四边形）。一个自然的问题是试图进一步将结果推广到具有更多边的循环多边形。令人惊讶的是，事实证明这远非简单，并且对于具有\（n> 4 \）的循环多边形不存在任何明确的解决方案。边缘。在本文中，我们基于一种新的基本方法来研究此类问题，其基础是苍鹭和Brahmagupta等式所基于的简单几何图形隐藏了游戏的真正参与者。详细地讲，我们建议重点关注由适当的三角形多边形的内切圆的内切线确定的边缘的剖切，这表明该方法可得出面积的明确公式，作为这些线段长度的对称函数。我们还表明，这种对称性可以在Heron和Brahmagupta的结果中重新发现，因此代表了所提供的一般等式的特殊情况。