We analytically characterize equilateral triangles. Our characterization includes the classically well-known characterizations: for a given triangle in the Euclidean plane, if the centroid and incenter of the triangle coincide, then the triangle must be equilateral; for a given triangle in the Euclidean plane, if the centroid and circumcenter of the triangle coincide, then the triangle must be equilateral. In our characterization, we consider the convolution of a radially symmetric function and the characteristic function of a triangle. The centroid, incenter and circumcenter of a triangle are described in terms of critical points of such a convolution.

中文翻译：

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等边三角形的解析表征

我们分析性地刻画了等边三角形。我们的表征包括经典的众所周知的表征：对于在欧几里得平面中的给定三角形，如果三角形的质心和中心重合，则三角形必须等边；对于欧几里得平面中的给定三角形，如果三角形的质心和外心重合，则三角形必须是等边的。在表征中，我们考虑了径向对称函数和三角形特征函数的卷积。根据这种卷积的临界点描述了三角形的质心，内心和外心。