AI 中文总结
本文研究三角形内一点处弦的平均长度，给出初等公式，并证明对所有三角形及内部点，平均弦长不超过周长的特定倍数，等号仅当三角形为等边且点为中心时成立。
AI 中文摘要
设$P$为三角形$T$内的一点。我们考虑通过$P$的弦的平均长度，其中弦的方向均匀选取。我们得到了一个初等公式，该公式用$P$到三角形各边和顶点的距离表示。几个经典三角形中心给出了特别简单的特例。例如，若$I$为内心，则\\[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \\] 我们的主要结果是尖锐不等式\\[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \\] 该不等式对所有周长为$p$的三角形和所有内部点$P$同时成立。因此，在所有这样的对$(T,P)$中，最大的平均弦长仅在$T$为等边三角形且$P$为其中心时取得。证明是一个初等对称化论证。最后，我们简要评述了该问题与凸体的径向中心、三角形的静电势中心以及对偶拟质量积分的关系。
英文摘要
Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}π \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{π\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.
