三角形中辅助线的作法
一．有关中线，中点。
题目中涉及到中线，中点。应想到以下几点1.倍长中线2.直角三角形的中线3.等腰三角形的三线合一 4.中线把面积分为相等的两部分 5.中位线
1．如图，已知ΔABC中，AB=5，AC=3，连BC上的中线AD=2，求BC的长。
![](https://wkrtcs.bdimg.com/rtcs/image?w=162.06666666667&md5sum=941e7925171502739ecb1900c97dc698&sign=ecb18fd2cb&rtcs_flag=1&rtcs_ver=3.13&l=webapp&bucketNum=73&ipr={"t":"img","w":"162.06666666667","h":"115.2","dataType":"png","c":"word/media/image1.png","imgOriW":308,"imgOriH":153})
2．如图，已知ΔABC中，AD是∠BAC的平分线，AD又是BC边上的中线。求证：ΔABC是等![](https://wkrtcs.bdimg.com/rtcs/image?w=93.666666666667&md5sum=941e7925171502739ecb1900c97dc698&sign=ecb18fd2cb&rtcs_flag=1&rtcs_ver=3.13&l=webapp&bucketNum=73&ipr={"t":"img","w":"93.666666666667","h":"159.33333333333","dataType":"png","c":"word/media/image2.png","imgOriW":91,"imgOriH":174})
腰三角形。
![](https://wkrtcs.bdimg.com/rtcs/image?w=174.66666666667&md5sum=941e7925171502739ecb1900c97dc698&sign=ecb18fd2cb&rtcs_flag=1&rtcs_ver=3.13&l=webapp&bucketNum=73&ipr={"t":"img","w":"174.66666666667","h":"234.66666666667","dataType":"gif","c":"word/media/image4.png"})
3.如图，AD为△ABC的中线，且∠1=∠2，∠3=∠4，求证：BE+CF>EF。
4.如图，AD为△ABC的中线，求证：AB+AC>2AD。
![](https://wkrtcs.bdimg.com/rtcs/image?w=156&md5sum=941e7925171502739ecb1900c97dc698&sign=ecb18fd2cb&rtcs_flag=1&rtcs_ver=3.13&l=webapp&bucketNum=73&ipr={"t":"img","w":"156","h":"208","dataType":"gif","c":"word/media/image5.png"})
5.如图，AB=6，AC=8，D为BC 的中点，求AD的取值范围。
![](https://wkrtcs.bdimg.com/rtcs/image?w=173.33333333333&md5sum=941e7925171502739ecb1900c97dc698&sign=ecb18fd2cb&rtcs_flag=1&rtcs_ver=3.13&l=webapp&bucketNum=73&ipr={"t":"img","w":"173.33333333333","h":"122.66666666667","dataType":"gif","c":"word/media/image6.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=190&md5sum=941e7925171502739ecb1900c97dc698&sign=ecb18fd2cb&rtcs_flag=1&rtcs_ver=3.13&l=webapp&bucketNum=73&ipr={"t":"img","w":"190","h":"147","dataType":"png","c":"word/media/image7_1.png","imgOriW":191,"imgOriH":148})
6.如图，△ABC中，E、F分别在AB、AC上，DE⊥DF，D是中点，试比较BE+CF与EF的大小.
7．如图，已知梯形ABCD中，AB//DC，AC⊥BC，AD⊥BD，求证：AC=BD。
![](https://wkrtcs.bdimg.com/rtcs/image?w=145.26666666667&md5sum=941e7925171502739ecb1900c97dc698&sign=ecb18fd2cb&rtcs_flag=1&rtcs_ver=3.13&l=webapp&bucketNum=73&ipr={"t":"img","w":"145.26666666667","h":"112.46666666667","dataType":"png","c":"word/media/image8.png","imgOriW":116,"imgOriH":106})
8.已知，如图，∠C=2∠A，AC=2BC。求证：△ABC是直角三角形。
![](https://wkrtcs.bdimg.com/rtcs/image?w=182.66666666667&md5sum=941e7925171502739ecb1900c97dc698&sign=ecb18fd2cb&rtcs_flag=1&rtcs_ver=3.13&l=webapp&bucketNum=73&ipr={"t":"img","w":"182.66666666667","h":"124","dataType":"gif","c":"word/media/image9.png"})
9．已知：如图，AB=2AC，∠1=∠2，DA=DB，求证：DC⊥AC
![](https://wkrtcs.bdimg.com/rtcs/image?w=157.33333333333&md5sum=941e7925171502739ecb1900c97dc698&sign=ecb18fd2cb&rtcs_flag=1&rtcs_ver=3.13&l=webapp&bucketNum=73&ipr={"t":"img","w":"157.33333333333","h":"142.66666666667","dataType":"gif","c":"word/media/image10.png"})
10．如图，ΔABC是等腰直角三角形，∠BAC=90°，BD平分∠ABC交AC于点D，CE垂![](https://wkrtcs.bdimg.com/rtcs/image?w=108.66666666667&md5sum=941e7925171502739ecb1900c97dc698&sign=ecb18fd2cb&rtcs_flag=1&rtcs_ver=3.13&l=webapp&bucketNum=73&ipr={"t":"img","w":"108.66666666667","h":"139.26666666667","dataType":"png","c":"word/media/image11.png","imgOriW":117,"imgOriH":166})
直于BD，交BD的延长线于点E。求证：BD=2CE。
11．如图，ΔABC中，AD是中线，延长AD到E，使DE=AD，DF是ΔDCE的中线。已知ΔABC的面积为2，求：ΔCDF的面积。
![](https://wkrtcs.bdimg.com/rtcs/image?w=158.2&md5sum=941e7925171502739ecb1900c97dc698&sign=ecb18fd2cb&rtcs_flag=1&rtcs_ver=3.13&l=webapp&bucketNum=73&ipr={"t":"img","w":"158.2","h":"135.2","dataType":"png","c":"word/media/image12.png","imgOriW":300,"imgOriH":151})
12．如图，在四边形ABCD中，AB=CD，E、F分别是BC、AD的中点，BA、CD的延长线分别交EF的延长线G、H。求证：∠BGE=∠CHE。
![](https://wkrtcs.bdimg.com/rtcs/image?w=141.33333333333&md5sum=941e7925171502739ecb1900c97dc698&sign=ecb18fd2cb&rtcs_flag=1&rtcs_ver=3.13&l=webapp&bucketNum=73&ipr={"t":"img","w":"141.33333333333","h":"149","dataType":"png","c":"word/media/image1.png","imgOriW":308,"imgOriH":153})
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