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高中数学竞赛教程 平面几何

高中数学竞赛教程 平面几何

DIYIJIANG ZHUYITIANJIAPINGXINGXIANZHENGTI

棋牌游戏ZAITONGYIPINGMIANNEI,BUXIANGJIAODELIANGTIAOZHIXIANJIAOPINGXINGXIAN.PINGXINGXIANSHICHUZHONGPINGMIANJIHEZUIJIBENDE,YESHIFEICHANGZHONGYAODETUXING.ZAIZHENGMINGMOUXIEPINGMIANJIHEWENTISHI,RUONENGYIJUZHENGTIDEXUYAO,TIANJIAQIADANGDEPINGXINGXIAN,ZENENGSHIZHENGMINGSHUNCHANG、JIANJIE. TIANJIAPINGXINGXIANZHENGTI,YIBANYOURUXIASIZHONGQINGKUANG.

1 WEILEGAIBIANJIAODEWEIZHI

DAJIAZHIDAO,LIANGTIAOPINGXINGZHIXIANBEIDISANTIAOZHIXIANSUOJIE,TONGWEIJIAOXIANGDENG,NEICUOJIAOXIANGDENG,TONGPANGNEIJIAOHUBU.LIYONGZHEIXIEXINGZHI,CHANGKETONGGUOTIANJIAPINGXINGXIAN,JIANGMOUXIEJIAODEWEIZHIGAIBIAN,YIMANZUQIUJIEDEXUYAO.

LI1 SHEP、QWEIXIANDUANBCSHANGLIANGDIAN,QIEBP=CQ,AWEIBCWAIYIDONGDIAN(RUTU1).DANGDIANAYUNDONGDAOSHI ∠BAP=∠CAQSHI,△ABCSHISHENMESANJIAOXING?SHIZHENGMINGNIDEJIELUN. DA: DANGDIANAYUNDONGDAOSHI∠BAP=∠CAQSHI,△ABCWEIDENGYAOSANJIAOXING.

ZHENGMING:RUTU1,FENBIEGUODIANP、BZUOAC、AQDEPINGXINGXIANDEJIAODIAND.LIANJIEDA.

ZAI△DBP=∠AQCZHONG,XIANRAN

∠DBP=∠AQC,∠DPB=∠C.

D

A

BPTU1

Q

棋牌游戏YOUBP=CQ,KEZHI △DBP≌△AQC. YOUDP=AC,∠BDP=∠QAC. YUSHI,DA∥BP,∠BAP=∠BDP.

ZEA、D、B、PSIDIANGONGYUAN,QIESIBIANXINGADBPWEIDENGYAOTIXING.GUAB=DP. SUOYIAB=AC.

ZHEILI,TONGGUOZUOPINGXINGXIAN,JIANG∠QAC“PINGTUI”DAO∠BDPDEWEIZHI.YOUYUA、D、B、PSIDIANGONGYUAN,SHIZHENGMINGHENSHUNCHANG. LI2 RUTU2,SIBIANXINGABCDWEIPINGXINGSIBIANXING,∠BAF=∠BCE.QIUZHENG:∠EBA=∠ADE. EZHENGMING:RUTU2,FENBIEGUODIANA、BZUOED、ECDEPINGXINGXIAN,DEJIAODIANP,LIANPE. YOUAB ∥ CD,YIZHI△PBA≌△ECD.YOUPA=ED,PB=EC. =

棋牌游戏XIANRAN,SIBIANXINGPBCE、PADEJUNWEIPINGXINGSIBIANXING.YOU ∠BCE=∠BPE,∠APE=∠ADE. YOU∠BAF=∠BCE,KEZHI ∠BAF=∠BPE.

棋牌游戏YOUP、B、A、ESIDIANGONGYUAN. YUSHI,∠EBA=∠APE. SUOYI,∠EBA=∠ADE.

P

BTU2

DC

F

ZHEILI,TONGGUOTIANJIAPINGXINGXIAN,SHIYIZHIYUWEIZHIZHONGDESIGEJIAOTONGGUOP、B、A、ESIDIANGONGYUAN,JINMILIANXIQILAI.∠APECHENGWEI∠EBAYU∠ADEXIANGDENGDEMEIJIE,ZHENGFAHENQIAOMIAO. 2 YU“SONG”XIANDUANDAODANGCHU

棋牌游戏LIYONG“PINGXINGXIANJIANJULIXIANGDENG”、“JIAZAIPINGXINGXIANJIANDEPINGXINGXIANDUANXIANGDENG”ZHEILIANGTIAO,CHANGKETONGGUOTIANJIAPINGXINGXIAN,JIANGMOUXIEXIANDUAN“SONG”DAOQIADANGWEIZHI,YIZHENGTI.

LI3 ZAI△ABCZHONG,BD、CEWEIJIAOPINGFENXIAN,PWEIEDSHANGRENYIYIDIAN.GUOPFENBIEZUOAC、AB、BCDECHUIXIAN,M、N、QWEICHUIZU.QIUZHENG:PM+PN=PQ.

A

ZHENGMING:RUTU3,GUODIANPZUOABDEPINGXINGXIANJIAOBDYUF,GUODIANFZUOBCDEPINGXINGXIANFENBIEJIAOPQ、AC NM

E

YUK、G,LIANPG. D YOUBDPINGXING∠ABC,KEZHIDIANFDAOAB、BCLIANGBIANJULIXIANGDENG.YOUKQ=PN. XIANRAN,

EPPD

B

K

QTU3

C

EFFD

CGGD

,KEZHIPG∥EC.

YOUCEPINGFEN∠BCA,ZHIGPPINGFEN∠FGA.YOUPK=PM.YUSHI, PM+PN=PK+KQ=PQ.

ZHEILI,TONGGUOTIANJIAPINGXINGXIAN,JIANGPQ“QIAKAI”CHENGLIANGDUAN,ZHENGDEPM=PK,JIUYOUPM+PN=PQ.ZHENGFAFEICHANGJIANJIE. 3 WEILEXIANDUANBIDEZHUANHUA

棋牌游戏YOUYU“PINGXINGYUSANJIAOXINGYIBIANDEZHIXIANJIEQITALIANGBIAN,SUODEDUIYINGXIANDUANCHENGBILI”,ZAIYIXIEWENTIZHONG,KEYITONGGUOTIANJIAPINGXINGXIAN,SHIXIANMOUXIEXIANDUANBIDELIANGXINGZHUANHUA.ZHEIZAIPINGMIANJIHEZHENGTIZHONGSHIHUIJINGCHANGYUDAODE.

LI4 SHEM1、M2SHI△ABCDEBCBIANSHANGDEDIAN,QIEBM1=CM2.RENZUOYIZHIXIANFENBIEJIAOAB、AC、AM1、AM2YUP、Q、N1、N2.SHIZHENG:

ABAP

ACAQ

AM

1

AN1

AMAN

22

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