1. **Problem Statement:** We are given a quadrilateral with points A, B, C, D, E, F such that AD = DB and AF = FC. The lengths DF = 14 and DE = 10 are known. We need to find the length of segment BC. 2. **Understanding the Problem:** Since AD = DB, point D is the midpoint of segment AB. Similarly, AF = FC means point F is the midpoint of segment AC. 3. **Key Insight:** If D and F are midpoints of AB and AC respectively, then segment DF is a mid-segment in triangle ABC. By the Mid-segment Theorem, segment DF is parallel to BC and its length is half of BC. 4. **Using the Mid-segment Theorem:** $$DF = \frac{1}{2} BC$$ Given $$DF = 14$$, so $$14 = \frac{1}{2} BC$$ 5. **Solving for BC:** Multiply both sides by 2: $$BC = 2 \times 14 = 28$$ 6. **Conclusion:** The length of segment BC is 28 units. Note: The length DE = 10 is extra information not needed to find BC in this problem.