题目内容
如图.△ABC是圆的内接正三角形,D是BC弧上任意一点,试探求线段BD、DC、DE之间的关系并予证明.![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103200058013593698/SYS201311032000580135936024_ST/images0.png)
【答案】分析:延长CD至H,使DH=BD,△ABC为正三角形,然后可证△ABD≌△CBH,得BD+DC=HC=AD,利用(同弧所对的圆周角相等)求△ABD∽△CED,然后利用其对应边成比例即可证明.
解答:
解:
+
=![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103200058013593698/SYS201311032000580135936024_DA/2.png)
证明:延长CD至H,使DH=BD,△ABC为正三角形,
∴∠BDC=120°,∠BDH=180°-∠BDC=60°,
∴△DBH为正三角形,
∴BD=HD,∠H=60°,
又∵AB=BC,∠ADB=∠ACB=60°,∠BAD=∠BCD,
∴△ABD≌△CBH,
∴AD=HC=HD+DC=BD+DC,
又∵∠DCE=∠BAD,∠EDC=∠ADB,(同弧所对的圆周角相等)
∴△ABD∽△CED
∴![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103200058013593698/SYS201311032000580135936024_DA/3.png)
∴
=
,
∴![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103200058013593698/SYS201311032000580135936024_DA/6.png)
∴
+
=
.
点评:此题考查学生对相似三角形的判定与性质,全等三角形的判定与性质,等边三角形的判定与性质,圆周角定理等知识点的理解和掌握,综合性较强,难度较大.
解答:
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103200058013593698/SYS201311032000580135936024_DA/images0.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103200058013593698/SYS201311032000580135936024_DA/0.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103200058013593698/SYS201311032000580135936024_DA/1.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103200058013593698/SYS201311032000580135936024_DA/2.png)
证明:延长CD至H,使DH=BD,△ABC为正三角形,
∴∠BDC=120°,∠BDH=180°-∠BDC=60°,
∴△DBH为正三角形,
∴BD=HD,∠H=60°,
又∵AB=BC,∠ADB=∠ACB=60°,∠BAD=∠BCD,
∴△ABD≌△CBH,
∴AD=HC=HD+DC=BD+DC,
又∵∠DCE=∠BAD,∠EDC=∠ADB,(同弧所对的圆周角相等)
∴△ABD∽△CED
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103200058013593698/SYS201311032000580135936024_DA/3.png)
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103200058013593698/SYS201311032000580135936024_DA/4.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103200058013593698/SYS201311032000580135936024_DA/5.png)
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103200058013593698/SYS201311032000580135936024_DA/6.png)
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103200058013593698/SYS201311032000580135936024_DA/7.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103200058013593698/SYS201311032000580135936024_DA/8.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103200058013593698/SYS201311032000580135936024_DA/9.png)
点评:此题考查学生对相似三角形的判定与性质,全等三角形的判定与性质,等边三角形的判定与性质,圆周角定理等知识点的理解和掌握,综合性较强,难度较大.
![](http://thumb2018.1010pic.com/images/loading.gif)
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