已知AABC是锐角三角形，过点A作ADLBC于点D，延长DA至点E，使DE=BC，点F在边AC上，连接DF，EF，使CDF=BAD，FD=AB.求证:FE=AC.---**Question Number:** 1. **Question Stem:** 如图, 已知 $\triangle ABC$ 是锐角三角形, 过点 $A$ 作 $AD \perp BC$ 于点 $D$, 延长 $DA$ 至点 $E$, 使 $DE=BC$, 点 $F$ 在边 $AC$ 上, 连接 $DF, EF$, 使 $\angle CDF = \angle BAD$, $FD=AB$. 求证: $FE=AC$. **Translation of Question Stem:** As shown in the figure, given that $\triangle ABC$ is an acute-angled triangle, draw $AD \perp BC$ from point $A$ to point $D$ on $BC$, extend $DA$ to point $E$ such that $DE=BC$, point $F$ is on side $AC$, connect $DF, EF$, such that $\angle CDF = \angle BAD$, $FD=AB$. Prove that: $FE=AC$. **Given Conditions:** 1. $\triangle ABC$ is an acute-angled triangle. 2. $AD \perp BC$ at point $D$. 3. $DA$ is extended to $E$ such that $DE=BC$. 4. Point $F$ is on side $AC$. 5. $\angle CDF = \angle BAD$. 6. $FD=AB$. **To Prove:** $FE=AC$. **Diagram Description:** * **Type:** Geometric figure, specifically a diagram illustrating a triangle and constructed points and segments. * **Elements:** * A triangle $ABC$ is shown. * Point $D$ is on the line segment $BC$. A perpendicular line segment $AD$ is drawn from vertex $A$ to $BC$. * The line segment $DA$ is extended upwards to a point $E$. * Point $F$ is on the line segment $AC$. * Line segments $DF$ and $EF$ are drawn. * **Labels:** * Vertices are labeled $A$, $B$, $C$, $E$. * Points are labeled $D$, $F$. * **Relative Positions and Connections:** * $A$ is above the line $BC$. $D$ is on the line $BC$. * $AD$ is a vertical line segment from $A$ to $D$. $AD \perp BC$ is indicated implicitly by the text. * $B, D, C$ appear to be roughly collinear, forming the base of $\triangle ABC$. $D$ is between $B$ and $C$. * $E$ is on the line passing through $A$ and $D$, such that $A$ is between $D$ and $E$. The segment $AE$ is shown extending upwards from $A$. The text states $DA$ is extended to $E$, meaning $D, A, E$ are collinear in that order, and $A$ is between $D$ and $E$. However, the diagram shows $A$ is between $D$ and $E$. Let's re-read the text. "延长 $DA$ 至点 $E$" means extend $DA$ to point $E$. This usually implies that $A$ is between $D$ and $E$. The diagram is consistent with this interpretation. * $F$ is on the line segment $AC$. * Segments $AB, BC, AC, AD, DF, EF$ are shown as lines connecting the respective points. * **Angles/Properties (Implied by text and diagram):** * $\angle ADB = 90^\circ$ (since $AD \perp BC$) * $\angle ADC = 90^\circ$ (since $AD \perp BC$) * $\angle CDF$ and $\angle BAD$ are marked in the text as being equal. * Lengths $DE=BC$ and $FD=AB$ are stated in the text. * Lengths $FE$ and $AC$ are stated in the text as needing to be proven equal.