1 . 如图一,在平面几何中,有如下命题“正三角形
的高为h,O是
内任意一点,则O到三边的距离的和为定值h,当O是
的中心时,O到各边的距离均为
”.
证明如下:设正三角形
边长为a,高h,O到三边的距离分别![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367c96a0ff95b92877eda2a7c98871e1.png)
则:
,即:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afca72943300013e376cabf814f241b7.png)
化简得,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64cb83e91782bf0f86cf6a8013ec480c.png)
若O是
中心,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcb93dd358c04249b78f9b4c023671c5.png)
即:正三角形中心到各边的距离均为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b265bcd9a636c4f85a5992438032db3.png)
![](https://img.xkw.com/dksih/QBM/2021/5/19/2724640426016768/2724926272692224/STEM/57cb2b31-c239-46c0-a316-169131e10e9e.png?resizew=209)
类比此命题及证明方法,在立体几何中,请写出高为h的正四面体
(图二)相应的命题,并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b265bcd9a636c4f85a5992438032db3.png)
证明如下:设正三角形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367c96a0ff95b92877eda2a7c98871e1.png)
则:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51fde79d55da7c6256f14cde7c555870.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afca72943300013e376cabf814f241b7.png)
化简得,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64cb83e91782bf0f86cf6a8013ec480c.png)
若O是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcb93dd358c04249b78f9b4c023671c5.png)
即:正三角形中心到各边的距离均为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b265bcd9a636c4f85a5992438032db3.png)
![](https://img.xkw.com/dksih/QBM/2021/5/19/2724640426016768/2724926272692224/STEM/57cb2b31-c239-46c0-a316-169131e10e9e.png?resizew=209)
类比此命题及证明方法,在立体几何中,请写出高为h的正四面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://img.xkw.com/dksih/QBM/2021/5/19/2724640426016768/2724926272692224/STEM/0553af6b-4ecc-442d-aa31-b9e87c7814d8.png?resizew=228)
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