根据附件图片内容，讲解“倍角模型”---Here is the extracted content from the image:
**Title:**
模型 51 “倍角” 模型 (Model 51 "Multiple Angle" Model)
**模型展现 (Model Presentation)**
Table Content:
| | Diagram Description | Diagram Description | Diagram Description |
| :---------- | :-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | :-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | :------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| 图示 (Diagram) | Triangle ABC. Vertices A, B, C. Sides AB, BC, AC. ∠B is approximately twice ∠C. | Triangle ABC. Point D on BC. Line segment AD is dashed. Vertices A, B, D, C are labeled. | Triangle ABC. Point D on the extension of CB. Line segment AD connects A to D. Vertices A, B, C, D are labeled. |
| 条件 (Condition) | 在△ABC中, ∠B=2∠C (In △ABC, ∠B=2∠C) | | |
| 作法 (Method) | 作2倍角的角平分线: 作∠B的平分线 BD (Draw the angle bisector of the double angle: Draw the angle bisector BD of ∠B) | 内作双等腰: 过点 A作∠DAC=∠C (Construct two isosceles triangles internally: From point A, draw ∠DAC=∠C) | 外作双等腰: 延长 CB 至点 D, 使得 BD=BA 作法: 内作双等腰: 过点 A作∠DAC=∠C (Construct two isosceles triangles externally: Extend CB to point D, such that BD=BA Method: Construct two isosceles triangles internally: From point A, draw ∠DAC=∠C) |
| 结论 (Conclusion) | △BCD 是等腰三角形 (△BCD is an isosceles triangle) | △ABD 与 △ADC 是等腰三角形 (△ABD and △ADC are isosceles triangles) | △ABD 与 △ADC 是等腰三角形 (△ABD and △ADC are isosceles triangles) |
QR Code: 视频讲模型 (Video explains model)
**思考延伸 (Extended Thinking)**
3 倍角模型: (3 Multiple Angle Model:)
条件: ∠BAC=3∠C (Condition: ∠BAC=3∠C)
图示: (Diagram:) Triangle ABC. Vertices A, B, C. Sides AB, BC, AC. ∠A is approximately three times ∠C.
**解题要点 (Problem Solving Point)**
当一个三角形中出现一个角是另一个角的2倍时, 可以通过作辅助线转化倍角寻找到等腰三角形解题.
(When one angle in a triangle is twice another angle, auxiliary lines can be drawn to transform the multiple angle and find isosceles triangles to solve the problem.)
**例 (Example)**
如图，在△ABC中，∠BAC=90°, D 为BC边上一点，连接 AD. 若 BD=3, AD=6, ∠ADC=2∠C, 则 CD 的长为____.
(As shown in the figure, in △ABC, ∠BAC=90°, D is a point on side BC, connect AD. If BD=3, AD=6, ∠ADC=2∠C, then the length of CD is ____.)
Model Reasoning:
模型猜想 存在角的倍数关系, 故 为“倍角”模型
(Model hypothesis: There is a multiple relationship between angles, therefore it is a "multiple angle" model)
Diagram: 例题图 (Example Problem Diagram)
Triangle ABC. ∠A is a right angle (90°). D is on BC. Line segment AD is drawn. Vertices A, B, C, D are labeled. AD appears to be a dashed line in this specific example diagram.
QR Code: 视频讲解 模型51例题 (Video explanation, Model 51 example problem)
答案见《答案详解详析》P90 (Answer can be found in "Detailed Answer Explanation" P90)
**模型解题三步法 (Model Problem Solving Three-Step Method)**
第一步: 依据特征找模型 (Step 1: Find the model based on characteristics)
特征: 同一个三角形中是 存在角度的倍数关系 三角形: △ADC 角度关系: ∠ADC=2∠C
(Characteristic: In the same triangle, there is a multiple relationship between angles. Triangle: △ADC. Angle relationship: ∠ADC=2∠C)
第二步: 抽象模型 (Step 2: Abstract the model)
Diagram: Triangle ADC. Vertices A, D, C are labeled. Sides AD, DC, AC.
第三步: 用模型 (Step 3: Apply the model)
内作双等腰: 过点A作∠EAC=∠C
(Construct two isosceles triangles internally: From point A, draw ∠EAC=∠C)
Diagram: Triangle ADC. Point E is on DC. Line segment AE is drawn (dashed). Vertices A, D, E, C are labeled.
Conclusion: △ADE与△AEC是等腰三角形 (△ADE and △AEC are isosceles triangles)