Capturing the hidden circle: An effective solution to middle school moving point maximum/minimum problems
抓取隐圆：初中动点最值问题的一种有效解法

摘要

动点最值问题是初中数学的高频难点，其核心障碍在于学生难以从动态图形中识别不变的几何关系。本文系统梳理了‘隐形圆’的三类基本模型：定点定长模型（动点到定点距离恒定，轨迹为圆）、定弦定角模型（动点对定线段所张角恒定，轨迹为圆弧）及四点共圆模型（对角互补或同底同侧等角）。结合三道典型例题，揭示了各自的关键构造方法——例1通过旋转构造全等三角形确定动点轨迹圆心；例2利用垂径定理和三角函数确定圆心位置与半径；例3通过全等三角形证得定角，再以四点共圆确定轨迹。在此基础上，提炼出“动中取定→定中找圆→圆中求最”的通用解题路径，并给出每步的操作要领。研究表明，该路径可使复杂的最值问题转化为定点到圆上动点的距离模型，大幅降低思维难度。教学上应借助几何画板等工具动态演示轨迹生成，帮助学生完成从“无圆”到“有圆”的认知跨越。

Abstract

Dynamic-point maximum/minimum problems constitute a frequent and difficult topic in junior high school mathematics. The core obstacle lies in students’ difficulty in identifying invariant geometric relationships from dynamic figures. This paper systematically sorts out three basic models of the “invisible circle”: the fixed-point-and-fixed-length model (the moving point maintains a constant distance from a fixed point, with a circle as its locus), the fixed-chord-and-fixed-angle model (the moving point subtends a constant angle to a fixed line segment, with a circular arc as its locus), and the concyclic-four-point model (characterized by supplementary opposite angles or equal angles on the same side of a common base).Combined with three typical examples, this paper reveals the key construction methods for each model. In Example 1, rotation is adopted to construct congruent triangles to locate the center of the moving-point locus. In Example 2, the perpendicular-chord theorem and trigonometric functions are applied to determine the center position and radius. In Example 3, congruent triangles are used to prove a fixed angle, and concyclicity of four points is then employed to confirm the locus. On this basis, a general problem-solving procedure of “identifying invariants from motion→locating the implicit circle from invariants→solving maximum/minimum values via point-circle distance” is summarized, with operational guidelines provided for each step. The study shows that this procedure transforms complex maximum/minimum problems into the distance model between a fixed point and a moving point on a circle, which greatly reduces cognitive load. From a teaching perspective, dynamic demonstration of locus generation with tools such as GeoGebra is recommended to help students achieve the cognitive leap from “no explicit circle” to “recognizing the implicit circle”.