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30 lines
991 B
30 lines
991 B
2 years ago
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**[前导知识:一线三垂直](https://zhuanlan.zhihu.com/p/346523929)**
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- 第一问是傻子题，不解释
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- $\triangle CPM$和$\triangle APN$面积相等，不好算，但都再加上四边形$MPNB$就好办多了：
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$\triangle CNB$与$\triangle AMB$面积相等，就好算了，因为这是两个直角三角形！
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$AB*BM=NB*BC$
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$5*t=8*(5-t)$
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$\therefore t=\frac{40}{13}$
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- 重点是第三问：
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- 我们发现，$t=3$时，$NB=3,AN=5$
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<font color='red' size=4><b>一线三垂直！！</b></font>
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从$N$引直线垂直于$CN$,
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从$A$引直线垂直于$AB$,
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两者交于$E$点
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$\triangle AEN \cong \triangle NBC$
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$\therefore EN=NC$
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连接$EC$,则 $\triangle ENC$是等腰直角三角形！
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$\therefore \angle NCE=45^{\circ}$
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同时，可以证明$\triangle AEC \cong \triangle ACM$
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$EC // AM$
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$\angle APN =\angle ECN=45^{\circ}$
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