三角形内心位于随机直径圆盘内的概率
The Probability That the Incenter of a Triangle Lies in a Random Diameter Disk
浏览论文内容
中文总结 AI 辅助
该研究针对三角形内心位于随机直径圆盘内的概率（即球面深度）问题，通过归一化锥测度的初等平面形式与转换原理推导出闭式公式，并证明了等边三角形时取最大值的严格不等式。
中文摘要 AI 辅助
设$P$和$Q$是从非退化三角形$ABC$内部均匀选取的独立点，$I$为该三角形的内心。我们研究以$PQ$为直径的闭圆盘包含$I$的概率。用多元统计学的语言来说，这就是$I$相对于三角形上均匀分布的球面深度（spherical depth）。\n 我们首先给出归一化锥测度构造的一种初等平面形式。若$O$是凸多边形的一个内点，则从$O$指向均匀分布内点的方向，与从$O$指向边界点的方向服从相同的分布，该边界点在每条边上的密度与$O$到对应边的距离成正比。由此可得，当且仅当该多边形是以内切圆圆心$O$为内心的切多边形时，这个边界点在弧长上是均匀的；对于三角形而言，这一性质刻画了内心的特征。\n 利用这一转换原理，我们得到闭式公式：\n\\[\mathrm{SphD}(I) =\left(\frac r s\right)^2 \left[ \frac{8R}{r}-1 -\Gamma(\cos A)-\Gamma(\cos B)-\Gamma(\cos C) \right], \\]\n其中$r$、$R$、$s$分别为内切圆半径、外接圆半径和半周长，且\n\\[ \Gamma(t)=\frac1t-\frac{1-t^2}{t^2}\mathrm{arctanh}\\ t \\]\n其连续取值为$\Gamma(0)=0$，$\Gamma(\pm1)=\pm1$。最后我们证明了严格不等式\n\\[\mathrm{SphD}(I)\le \frac13+\frac{\log 3}{6}, \\]\n当且仅当$ABC$为等边三角形时等号成立。
英文摘要
Let $P$ and $Q$ be independent points chosen uniformly from the interior of a nondegenerate triangle $ABC$, and let $I$ be its incenter. We study the probability that the closed disk with diameter $PQ$ contains $I$. In the language of multivariate statistics, this is the spherical depth of $I$ with respect to the uniform distribution on the triangle. We first give an elementary planar form of the normalized cone-measure construction. If $O$ is an interior point of a convex polygon, then the direction from $O$ to a uniformly distributed interior point has the same law as the direction from $O$ to a boundary point whose density on each side is proportional to the distance from $O$ to that side. Consequently, this boundary point is uniform in arclength if and only if the polygon is tangential with incircle center $O$; for a triangle, this characterizes the incenter. Using this transfer principle, we obtain the closed formula \[\mathrm{SphD}(I) =\left(\frac r s\right)^2 \left[ \frac{8R}{r}-1 -Γ(\cos A)-Γ(\cos B)-Γ(\cos C) \right], \] where $r,R,s$ are the inradius, circumradius, and semiperimeter, and \[ Γ(t)=\frac1t-\frac{1-t^2}{t^2}\mathrm{arctanh}\ t \] with continuous values $Γ(0)=0$ and $Γ(\pm1)=\pm1$. Finally we prove the sharp inequality \[\mathrm{SphD}(I)\le \frac13+\frac{\log 3}{6}, \] with equality if and only if $ABC$ is equilateral.
