一个……的几何题

最新推荐文章于 2025-07-18 08:56:56 发布

原创最新推荐文章于 2025-07-18 08:56:56 发布 · 274 阅读

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本文详细解析了一道涉及多个几何元素的复杂证明题，通过逐步推导展示了如何利用几何关系简化问题，并最终证明了两个关键等式的等价性。

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题目如下：
这里写图片描述
证明：

= = = = = AP \cdot DH HA \cdot PD - 2 ( AH + HP ) ( HP + PD ) AH \cdot PD - 2 AH \cdot HP - AH \cdot PD + HP 2 + HP \cdot PD AH \cdot PD ( AH + HP ) ( AH + HP + PD ) PD \cdot AH \cdot HP - AH \cdot PD + HP 2 + HP \cdot PD AH ( AH + HP ) ( AH + HP + PD ) AP \cdot AD PD \cdot (1 AH + HP + PD - 2 AH + HP + 1 AH) AP \cdot AD PD \cdot (1 AD + 1 AH - 2 AP) = 0

$\begin{align*} &\dfrac{\text{AP} \cdot\text{DH}}{\text{HA} \cdot\text{PD}}-2\\ ={}&\dfrac{(\text{AH}+\text{HP}) (\text{HP}+\text{PD})}{\text{AH} \cdot\text{PD}}-2\\ ={}&\dfrac{\text{AH} \cdot\text{HP}-\text{AH} \cdot\text{PD}+\text{HP}^2+\text{HP} \cdot\text{PD}}{\text{AH} \cdot \text{PD}}\\ ={}&\dfrac{(\text{AH}+\text{HP}) (\text{AH}+\text{HP}+\text{PD})}{\text{PD}}\cdot \dfrac{\text{AH} \cdot\text{HP}-\text{AH} \cdot\text{PD}+\text{HP}^2+\text{HP} \cdot\text{PD}}{\text{AH} (\text{AH}+\text{HP}) (\text{AH}+\text{HP}+\text{PD})}\\ ={}&\dfrac{\text{AP}\cdot \text{AD}}{\text{PD}}\cdot (\dfrac{1}{\text{AH}+\text{HP}+\text{PD}}-\dfrac{2}{\text{AH}+\text{HP}}+\dfrac{1}{\text{AH}})\\ ={}&\dfrac{\text{AP}\cdot \text{AD}}{\text{PD}}\cdot (\dfrac{1}{\text{AD}}+\dfrac{1}{\text{AH}}-\dfrac{2}{\text{AP}})=0 \end{align*}$

其实，就是：

1 AD + 1 AH - 2 AP = 0 和 AP \cdot DH HA \cdot PD - 2 = 0 等 价 于 AH \cdot HP - AH \cdot PD + HP 2 + HP \cdot PD = 0

$\dfrac{1}{\text{AD}}+\dfrac{1}{\text{AH}}-\dfrac{2}{\text{AP}}=0\text{和}\dfrac{\text{AP} \cdot\text{DH}}{\text{HA} \cdot\text{PD}}-2=0\text{等价于}\text{AH}\cdot \text{HP}-\text{AH}\cdot \text{PD}+\text{HP}^2+\text{HP} \cdot \text{PD}=0$