根据附件图片内容，解释”什么是阿氏圆模型“？---**Model Title:** 模型 54 “阿氏圆” 模型 (Model 54 "Apollonius Circle" Model) **Model Presentation:** * **Table Content:** * **Header:** 图示 (Diagram), 条件 (Conditions), 问题 (Question) * **Row 1:** * 条件: 点 P 是半径为 r 的 ⊙O 上一动点, 点 A, B 为 ⊙O 外两定点 (Point P is a moving point on ⊙O with radius r, points A and B are two fixed points outside ⊙O) * 问题: 当 r, k 满足 r = k ⋅ OA (0 < k < 1) 时, 如何确定点 P 的位置, 使得 kAP + BP 的值最小. (When r and k satisfy r = k ⋅ OA (0 < k < 1), how to determine the position of point P such that kAP + BP has the minimum value.) * **Diagram Description:** * Type: Geometric figure illustrating a circle and triangles. * Main Elements: * A circle with center O. * Points O, A, B, P. O is the center of the circle. A and B are fixed points outside the circle. P is a moving point on the circle. * Lines/Segments: OA, OB, AP, BP. * Labels: O, A, B, P. * Other: A QR code labeled "视频讲解" (Video Explanation) is present. **Conclusion Analysis:** * **Text:** * 如图, 点 P 是半径为 r 的 ⊙O 上一动点, 点 A, B 为 ⊙O 外的定点, 且 r = k ⋅ OA (0 < k < 1), 如何确定点 P 的位置, 使得 kAP + BP 的值最小. (As shown in the figure, point P is a moving point on ⊙O with radius r, points A and B are fixed points outside ⊙O, and r = k ⋅ OA (0 < k < 1). How to determine the position of point P such that kAP + BP has the minimum value.) * 一找: 找带有系数 k 的线段 AP; (Step 1: Find the line segment AP with coefficient k;) * 二构: 在线段 OA 上取一点 C, 构造 △PCO ∼ △APO; (Step 2: Construct: Take a point C on line segment OA, construct △PCO ∼ △APO;) * ① 在线段 OA 上截取 OC, 使 OC = k ⋅ r; (① Intercept OC on line segment OA such that OC = k ⋅ r;) * ② 连接 PC, OP, 证明 △PCO ∼ △APO; (② Connect PC, OP, prove △PCO ∼ △APO;) * 三转化: 通过相似三角形的对应边成比例, 将 kAP 转化为 PC; (Step 3: Transform: Using the ratio of corresponding sides of similar triangles, transform kAP into PC;) * 四求解: 使得 kAP + BP = PC + BP, 连接 BC, 利用“两点之间, 线段最短”转化为求 BC 的长. (Step 4: Solve: To make kAP + BP = PC + BP, connect BC, use the principle "the shortest distance between two points is a straight line" to transform the problem into finding the length of BC.) * **Diagram Description:** * Type: Geometric figure illustrating the construction steps for solving the problem. * Main Elements: * A circle with center O. * Points O, A, B, C, P, P'. O is the center. P is on the circle. A and B are outside the circle. C is on line segment OA. P' is shown above P. * Lines/Segments: OA, OB, AP, BP, OP, OC, PC, BC, OP'. * Shapes: Triangle PCO and triangle APO are depicted or implied by the labels and lines. **Thinking Extension:** * **Header:** ? 思考延伸 (Thinking Extension) * **Text:** * “阿氏圆”与“胡不归”之间的区别: (“Apollonius Circle” and “Hu Bu Gui” Differences:) * 1. 轨迹不同: 若点 P 的轨迹为一条直线, 则考虑“胡不归”模型; 若点 P 的轨迹为圆或圆的一部分, 则考虑“阿氏圆”模型; (1. Different locus: If the locus of point P is a straight line, then consider the "Hu Bu Gui" model; if the locus of point P is a circle or part of a circle, then consider the "Apollonius Circle" model;) * 2. 解题方法不同: “胡不归”模型是利用锐角三角形函数和垂线段最短解题; “阿氏圆”模型是利用“A 字”型相似三角形解题. (2. Different solving methods: The "Hu Bu Gui" model uses acute triangle trigonometric functions and the shortest perpendicular segment; the "Apollonius Circle" model uses "A-shaped" similar triangles to solve.) * **Header:** ? 解题要点 (Key points for solving) * **Text:** 当 kAP + BP 中系数 k 大于 1 时, 考虑提取 k 转化为 k(AP + 1/k BP), 然后按“阿氏圆”模型进行计算. (When the coefficient k in kAP + BP is greater than 1, consider factoring out k to transform it into k(AP + 1/k BP), and then calculate according to the "Apollonius Circle" model.) **Example 1:** * **Header:** 例 1 (Example 1) * **Question Stem:** * 如图, 在 △OAB 中, ∠AOB = 90°, OB = 4, OA = 6, ⊙O 的半径为 2, 点 P 为 ⊙O 上一动点, 则 AP + 1/2 BP 的最小值为 ______. (As shown in the figure, in △OAB, ∠AOB = 90°, OB = 4, OA = 6, the radius of ⊙O is 2, and point P is a moving point on ⊙O, then the minimum value of AP + 1/2 BP is ______.) * **Diagram Description:** * Type: Geometric figure illustrating a right triangle and a circle. * Main Elements: * Coordinate axes implied by the right angle at O. O is at the origin. * Point A on the positive Y-axis. * Point B on the positive X-axis. * A circle with center O and radius 2. * Point P is on the circle. * Lines/Segments: OA, OB, AB, OP, AP, BP. * Angle: ∠AOB = 90°. * Labels: O, A, B, P. * Data: OA = 6, OB = 4, radius of ⊙O = 2. * **Model Conjecture Text Box:** * 模型猜想: 存在平面上两定点, 圆上一动点, 求线段和最小值, 且一条线段带系数, 故为“阿氏圆”模型 (Model Conjecture: There are two fixed points on the plane, and a moving point on a circle. Find the minimum value of the sum of line segments, and one line segment has a coefficient, so it is the "Apollonius Circle" model) * **Answer Location:** * 答案见《答案详细解析》P96 (Answer can be found in "Detailed Answer Analysis" P96)