解答---**Extraction Content:** **Problem Title/Identifier:** 配套练习 3-1 **Question Stem:** 如图, 等边三角形 ABC 中, 放置等边三角形 DEF, 且点 D, E 分别落在 AB, BC 上, AD=5, 连接 CF, 若 CF 平分∠ACB, 则 BE 的长度为_______. **Mathematical Formulas/Equations/Conditions:** - Triangle ABC is equilateral. - Triangle DEF is equilateral. - Point D is on line segment AB. - Point E is on line segment BC. - AD = 5. - Line segment CF is drawn. - CF bisects angle ACB (∠ACB). **Chart/Diagram Description:** - Type: Geometric figure. - Main Elements: - Triangle ABC: A large triangle labeled with vertices A (top), B (bottom left), and C (bottom right). Appears to be an equilateral triangle as stated in the problem. - Triangle DEF: A smaller triangle placed inside triangle ABC. Vertex D is on side AB, vertex E is on side BC, and vertex F is inside triangle ABC. The triangle appears to be rotated relative to triangle ABC. - Points: A, B, C are vertices of the larger triangle. D is on AB, E is on BC, F is inside triangle ABC. - Line Segments: Sides of triangle ABC (AB, BC, AC), sides of triangle DEF (DE, EF, FD), segment AD and DB on AB, segment BE and EC on BC, and segment CF connecting point C to point F. - Labels: Points are labeled A, B, C, D, E, F. - Other Annotations: A handwritten 'S' is near point A. The length AD is given as 5. - Relative Position: D is on the upper part of AB. E is on the left part of BC. F is inside triangle ABC, roughly to the right of D and above E. CF is a line segment from C going upwards and leftwards towards F. **Other Relevant Text:** - (Implied from "equilateral triangle ABC") ∠ABC = ∠BCA = ∠CAB = 60°. - (Implied from "CF bisects ∠ACB") ∠BCF = ∠ACF = ∠ACB / 2 = 60° / 2 = 30°. - (Implied from "equilateral triangle DEF") DE = EF = FD and ∠EDF = ∠DFE = ∠FED = 60°.