根据附件图片内容，讲解从 动点轨迹为圆的主从联动 模型---Here is the extraction of the content from the image: **Title:** 能力点 动点轨迹为圆的“主从联动”模型 Ability Point: "Master-Slave Linkage" Model where the Locus of a Moving Point is a Circle **Method Summary Table:** | | A, Q, P 三点共线 (A, Q, P collinear) | A, Q, P 三点不共线 (A, Q, P non-collinear) | | :---------- | :----------------------------------------------------------------------------------------------------- | :--------------------------------------------------------------------------------------------------------- | | 特点 (Features) | 定点 A,动点 P 和 Q,∠PAQ=α, AP/AQ 为定值 k, 点 P 在 ⊙O 上运动 (Fixed point A, moving points P and Q, ∠PAQ=α, AP/AQ = constant k, point P moves on ⊙O) | 定点 A,动点 P 和 Q,∠PAQ=α, AP/AQ 为定值 k, 点 P 在 ⊙O 上运动 (Fixed point A, moving points P and Q, ∠PAQ=α, AP/AQ = constant k, point P moves on ⊙O) | | 图示 (Diagrams) | *Geometric Diagram 1:* Points A, Q, P, M are on a line. O is a point. Point P is on a circle centered at O. A dashed circle is shown, potentially related to M or Q. An arrow points to Diagram 2. | *Geometric Diagram 2:* Points A, Q, P form triangle APQ with angle ∠PAQ = α at A. Point P is on a circle centered at O. Point M is shown. A dashed circle is shown centered at M. Triangle AOM is shown. An arrow points from Diagram 1 to Diagram 2. | **Conclusion (结论):** When point P's locus is on a circle, point Q's locus is on a circle: ① The angle between the line connecting the two centers and the fixed point is equal to the angle between the lines connecting the master and slave points to the fixed point, i.e., ∠OAM = ∠PAQ; ② The ratio of the distances of the master and slave points to the fixed point is equal to the ratio of the distances of the two centers to the fixed point, and is also equal to the ratio of the radii of the two circles, i.e., AP:AQ=AO:AM=PO:QM=k:1 --- **Problem 1:** **(定点与两动点不共线)如图, AB=6,点 O 在线段 AB 上, AO=2, C 为平面内一点, OC=1, 连接 BC, 以 BC 为边在 BC 的上方作等边△BCD, 连接 AD,在图中分别作出点 C,D 的运动轨迹, 并求出 AD 的最小值为_______** (Fixed point and two moving points are not collinear) As shown in the figure, AB=6, point O is on line segment AB, AO=2, C is a point in the plane, OC=1, connect BC, construct an equilateral triangle △BCD above BC with BC as a side, connect AD, respectively draw the loci of points C and D in the figure, and find the minimum value of AD _______ **Diagram for Problem 1:** 1 题图 (Diagram for Problem 1) *Geometric Diagram:* Points A, O, B are collinear on a line segment AB, with O between A and B. Points C and D form triangles ABC and BCD. △BCD is indicated as an equilateral triangle. Line segments AO, OB, OC, BC, CD, DB, AC, AD are shown. --- **Key Point Guidance (关键点点拨):** Determination of the locus circle for two moving points and a fixed point with equal line segments when they are not collinear: ① Determine the center: Connect the fixed point and the center, combine known angles and line segments to determine the center of the locus of the slave point; ② Determine the radius: By constructing congruent triangles, obtain a fixed length and determine the radius length.