1 . 【教材呈现】下图是华师版八年级上册数学教材第96页的部分内容.
我们已经知道角是轴对称图形,角平分线所在的直线是角的对称轴.如图所示,
是
的平分线,P是
上任一点,作
,
,垂足分别为点D和点E.将
沿
对折,我们发现
与
完全重合.由此即有:角平分线的性质定理角平分线上的点到角两边的距离相等.
已知:如图所示,
是
的平分线,点P是
上的任意一点,
,
,垂足分别为点D和点E.
求证:
.
和
,只要证明这两个三角形全等,便可证得
.
(1)请根据教材中的分析,结合图①,写出“角平分线的性质定理”完整的证明过程.
【定理应用】
(2)如图②,已知
是
的平分线,点P是
上的任意一点,点D、E分别在边
上,连结
,
.若
,
,则
的长为______.
(3)如图③,在平行四边形
中,
,
平分
交
于点E,连结
,将
绕点E旋转,当点C的对应点F落在边
上时,若
,则四边形
的面积为______.
我们已经知道角是轴对称图形,角平分线所在的直线是角的对称轴.如图所示,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d7b2fe01a33c4825f9974ed9663a99c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3224457d394d71df7d2955be6149dea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c70aa83794b41e477e7b546ecbe435ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d7b2fe01a33c4825f9974ed9663a99c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc5adb5eb60ae4435a12d93854066298.png)
已知:如图所示,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d7b2fe01a33c4825f9974ed9663a99c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3224457d394d71df7d2955be6149dea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c70aa83794b41e477e7b546ecbe435ed.png)
求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/374e2198b495b86b0f8308d28035a3db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bdac31e82ed00eace31e8c075c97bb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e26e65f15f56505cea3672e8cd3648c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/374e2198b495b86b0f8308d28035a3db.png)
(1)请根据教材中的分析,结合图①,写出“角平分线的性质定理”完整的证明过程.
【定理应用】
(2)如图②,已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d7b2fe01a33c4825f9974ed9663a99c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/339773a4e6e557f0846ebcb5830145e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35a3cee81392974e0b90d98e4411380e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4f290de3719b2dcce153c947df94958.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc1eb76fe74cba30f7cbcde349ba80da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c4222e6d812bd93a212035b5528145a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
(3)如图③,在平行四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e075468e7fb0bf30229aec01a7205977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f121ec32ab7e72ef4734413fbbf1c3d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b312dab930cbbb9a4bb1a99f044dab73.png)
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