关于广州市越秀区2024九上数学期末考25的解析

题目描述

如图，四边形$ABCD$中，$AB= CD$，$\angle ABC+ \angle BCD= 270°$

（1）求$\angle A+ \angle D$
（2）连接$AC$,若$\angle ACB= 45°$,求证： $BC^{2} +2AC^{2} =AD^{2}$
（3）点$E$，$F$分别为线段$BC$和$AD$上的点，点$G$是线段$EF$上任意一点且$\triangle GAB$和$\triangle GCD$的面积相等，过点$D$作$DH \perp EF$，$DH$交直线$EF$于点$H$，连接$AH$。若$AD= 4$，求$AH_{min}$

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

## 第一问 瞪眼法秒了，$90°$

第二问

既然$AB= CD$，而且$\angle BAD+ \angle D= 90°$，那么我们不妨把$\triangle ACD$挪走。

• 辅助线①号：把$AD$绕点$A$逆时针旋转$90°$得$AF$，连接$BF$
  这时，易证：$\triangle A D C \cong \triangle A B F$

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

然后呢？ 图里有一个等腰！（$\triangle ADF$） 一个等腰，一个$45°$，手拉手出来了吧。

• 辅助线②号：把$C$点绕$A$逆时针旋转$90°$得到点$G$，连$GC$，$AG$，$GF$，$BG$与$AF$交于$E$
  此时，易证：$G$，$B$，$C$共线；$\triangle A D C \cong \triangle A F G$

这时有▱$ABFG$。

然后开始设元暴证！

• 辅助线③号：作$AI \perp GC$于$I$
  设$GE= EB= a$，$BC= b$

$$GI= IC= \frac{2a+b}{2}$$

$$AC= \frac{\sqrt{2} (2a+b)}{2}$$

$$AD= AF= 2AE= 2\sqrt{IE^2+ AI^2}= ...= \frac{4}{4}$$

$$BC^2+ 2AC^2= b^2+ (2a+ b)^2=$$