用中国北师大教材初一下学期的思维---23. (本小题 13 分)
【新考向·过程学习】【问题初探】
在数学活动课上, 李老师给出如下问题: 如图 1, 在△ABC (AB ≠ AC) 中, 点 D, E 在边 BC 上, DE = CE, 过点 D 作 DF // AB 交 AE 于点 F, 且 DF = AC. 判断 AE 是否平分 ∠BAC, 并说明理由.
下面是两位同学的做法:
小美同学从线段 FE 的角度去思考, 如图 2, 延长 FE 至点 G, 使 EG = EF, 连接 CG;
小丽同学从线段 AE 的角度去考虑, 如图 3, 延长 AE 至点 G, 使 EG = EA, 连接 DG.
(1) 请你选择一名同学的解题思路, 写出证明过程;
【类比分析】
(2) 如图 4, 在△ABC 中, BD = AC, CE 是△ACD 的中线, ∠ACD = ∠ADC. 请判断 BC 与 CE 之间的数量关系, 并说明理由;
【学以致用】
(3) 如图 5, 在△ABC 中, 分别以 AB, AC 为直角边向内作等腰直角三角形, ∠BAE = ∠CAF = 90°, AE = AB, AC = AF, AD 是 BC 边上的中线, 且 AD = 8, 求 EF 的长.
更多“过程学习”试题见 P78-P80
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Chart/Diagram Description:
**图 1:**
- Type: Geometric figure (Triangle ABC with points on one side and an intersecting segment).
- Main Elements:
- Vertices: A, B, C of a triangle.
- Points: D and E on the side BC. F on the segment AE.
- Line segments: BC, DE, CE, AE, DF, AF, FE.
- Relationships: D and E are on BC. DF is drawn from D, parallel to AB, intersecting AE at F.
- Labels: A, B, C, D, E, F. "第 23 题图 1".
**图 2:**
- Type: Geometric figure (Triangle ABC with points on one side and extended segments).
- Main Elements:
- Vertices: A, B, C of a triangle.
- Points: D and E on BC. F on AE. G is an extension of FE.
- Line segments: BC, DE, CE, AE, DF, FE, EG, CG.
- Relationships: D and E on BC. DF intersects AE at F. FE is extended through E to G such that EG = EF. CG is connected (shown as a dashed line).
- Labels: A, B, C, D, E, F, G. "第 23 题图 2".
**图 3:**
- Type: Geometric figure (Triangle ABC with points on one side and extended segments).
- Main Elements:
- Vertices: A, B, C of a triangle.
- Points: D and E on BC. F on AE. G is an extension of AE.
- Line segments: BC, DE, CE, AE, DF, FE, EG, DG.
- Relationships: D and E on BC. DF intersects AE at F. AE is extended through E to G such that EG = EA. DG is connected (shown as a dashed line).
- Labels: A, B, C, D, E, F, G. "第 23 题图 3".
**图 4:**
- Type: Geometric figure (Triangle ABC with a point on one side and a median).
- Main Elements:
- Vertices: A, B, C of a triangle.
- Point: D on BC. E on AD.
- Line segments: BC, BD, CD, AC, AD, CE, BE (implied connection).
- Relationships: D is on BC. E is the midpoint of AD (since CE is the median of △ACD).
- Labels: A, B, C, D, E. "第 23 题图 4".
**图 5:**
- Type: Geometric figure (Triangle ABC with internal isosceles right-angled triangles and a median).
- Main Elements:
- Vertices: A, B, C of a triangle.
- Points: E and F internal to the triangle formed by vertices A, B, C. D is on BC.
- Line segments: AB, AC, AE, AF, BC, AD, BD, CD, EF.
- Relationships: △ABE is an isosceles right-angled triangle with right angle at A (implied by ∠BAE=90° and AE=AB). △ACF is an isosceles right-angled triangle with right angle at A (implied by ∠CAF=90° and AC=AF). E and F are positioned such that ∠BAE=90° and ∠CAF=90°. D is the midpoint of BC.
- Labels: A, B, C, D, E, F. "第 23 题图 5".