（2012?柳州）如图，AB是⊙O的直径，AC是弦．（1）请你按下面步骤画图（画图或作辅助线时先使用铅笔画出 如图，AB是⊙O的直径，AC是弦． （1）请你按下面步骤画...
\u5982\u56fe\uff0cAB\u662f\u2299O\u7684\u76f4\u5f84\uff0cAC\u662f\u5f26\uff0e \uff081\uff09\u8bf7\u4f60\u6309\u4e0b\u9762\u6b65\u9aa4\u753b\u56fe\uff08\u753b\u56fe\u6216\u4f5c\u8f85\u52a9\u7ebf\u65f6\u5148\u4f7f\u7528\u94c5\u7b14\u753b\u51fa\uff0c\u786e\u5b9a\u540e\u5fc5\u987b\uff081\uff09\u5982\u4e0b\u56fe\uff1b\uff082\uff09\u8be6\u89c1\u89e3\u6790\uff1b\uff083\uff09 \u8bd5\u9898\u5206\u6790\uff1a\uff081\uff09\u6839\u636e\u57fa\u672c\u4f5c\u56fe\u4f5c\u51fa\u2220BAC\u7684\u89d2\u5e73\u5206\u7ebfAD\u4ea4\u2299O\u4e8e\u70b9D\uff1b\u70b9D\u4f5cAC\u7684\u5782\u7ebf\uff0c\u5782\u8db3\u4e3a\u70b9E\uff1b\uff082\uff09\u8fde\u63a5OD\uff0c\u5148\u6839\u636e\u5706\u7684\u57fa\u672c\u6027\u8d28\u53ef\u5f97OA=OD\uff0c\u6839\u636e\u201c\u7b49\u8fb9\u5bf9\u7b49\u89d2\u201d\u53ef\u5f97\u2220OAD=\u2220ODA\uff0c\u518d\u7ed3\u5408AD\u5e73\u5206\u2220BAC\u53ef\u5f97\u2220EAD=\u2220ODA\uff0c\u5373\u53ef\u8bc1\u5f97AC\u2225OD\uff0c\u7136\u540e\u6839\u636e\u5e73\u884c\u7ebf\u7684\u6027\u8d28\u5373\u53ef\u4f5c\u51fa\u5224\u65ad\uff1b\uff083\uff09\u6839\u636e\u76f4\u5f84\u6240\u5bf9\u7684\u5706\u5468\u89d2\u4e3a\u76f4\u89d2\u5f97\u5230\u2220ADB=90\u00b0\uff0c\u800cDE\u22a5AC\uff0c\u5219\u2220AED=90\u00b0\uff0c\u53c8\u7531AD\u5e73\u5206\u2220CAB\u5f97\u5230\u25b3AED\u223d\u25b3ADB\uff0c\u518d\u6839\u636e\u76f8\u4f3c\u4e09\u89d2\u5f62\u7684\u5bf9\u5e94\u8fb9\u6210\u6bd4\u4f8b\u5373\u53ef\u6c42\u5f97\u7ed3\u679c.\u8bd5\u9898\u89e3\u6790\uff1a\uff081\uff09\u5982\u56fe\u6240\u793a\uff1a \uff082\uff09\u8fde\u63a5OD \u2235OA=OD\u2234\u2220OAD=\u2220ODA\uff0c\u2235AD\u5e73\u5206\u2220BAC\u2234\u2220EAD=\u2220OAD\u2234\u2220EAD=\u2220ODA\u2234AC\u2225OD\u2235DE\u22a5AC\u2234\u2220EDO=90\u00b0\u2234DE\u662f\u2299O\u7684\u5207\u7ebf\uff1b\uff083\uff09\u2235AB\u662f\u2299O\u7684\u76f4\u5f84\uff0c\u2234\u2220ADB=90\u00b0\uff0c\u2235DE\u22a5AC\uff0c\u2234\u2220AED=90\u00b0\uff0c\u2235AD\u5e73\u5206\u2220CAB\uff0c\u2234\u2220CAD=\u2220DAB\uff0c\u2234Rt\u25b3ADE\u223dRt\u25b3ABD\uff0c \u2234 \u2235AD=5\uff0cAE=4\u2234 \uff0c\u89e3\u5f97 \uff0c\u5373\u2299O\u7684\u76f4\u5f84\u4e3a \uff0e
\uff081\uff09\u5982\u56fe\uff1b \uff082\uff09\u5148\u6839\u636e\u5706\u5468\u89d2\u5b9a\u7406\u53ef\u5f97\u2220ADB=90\u00b0\uff0c\u518d\u7ed3\u5408DE\u22a5AC\uff0cAD\u5e73\u5206\u2220CAB\uff0c\u5373\u53ef\u8bc1\u5f97Rt\u25b3ADE\u223dRt\u25b3ABD\uff0c\u6839\u636e\u76f8\u4f3c\u4e09\u89d2\u5f62\u7684\u6027\u8d28\u5373\u53ef\u5f97\u5230\u7ed3\u8bba\uff1b\uff083\uff09 \u8bd5\u9898\u5206\u6790\uff1a\uff081\uff09\u6839\u636e\u89d2\u5e73\u5206\u7ebf\u4e0e\u5782\u7ebf\u7684\u753b\u6cd5\u5373\u53ef\u4f5c\u51fa\u56fe\u5f62\uff1b\uff082\uff09\u5148\u6839\u636e\u5706\u5468\u89d2\u5b9a\u7406\u53ef\u5f97\u2220ADB=90\u00b0\uff0c\u518d\u7ed3\u5408DE\u22a5AC\uff0cAD\u5e73\u5206\u2220CAB\uff0c\u5373\u53ef\u8bc1\u5f97Rt\u25b3ADE\u223dRt\u25b3ABD\uff0c\u6839\u636e\u76f8\u4f3c\u4e09\u89d2\u5f62\u7684\u6027\u8d28\u5373\u53ef\u5f97\u5230\u7ed3\u8bba\uff1b\uff083\uff09\u8fdeOD\u3001BC\uff0c\u5b83\u4eec\u4ea4\u4e8e\u70b9G\uff0c\u75315AC=3AB\uff0c\u53ef\u8bbeAC=3x\uff0cAB=5x\uff0c\u6839\u636e\u5706\u5468\u89d2\u5b9a\u7406\u53ef\u5f97\u2220ACB=90\u00b0\uff0c\u7531\u2220CAD=\u2220DAB\uff0c\u53ef\u7684\u5f27DC=\u5f27DB\uff0c\u5373\u53ef\u5f97\u5230OD\u2225AE\uff0cOG= AC= \uff0c\u4ece\u800c\u8bc1\u5f97\u56db\u8fb9\u5f62ECGD\u4e3a\u77e9\u5f62\uff0c\u53ef\u7684CE=DG=OD-OG= x- x =x\uff0c\u5219AE=AC+CE=3x+x=4x\uff0c\u6839\u636eAE\u2225OD\uff0c\u53ef\u5f97\u25b3AEF\u223d\u25b3DOF\uff0c\u6839\u636e\u76f8\u4f3c\u4e09\u89d2\u5f62\u7684\u6027\u8d28\u5373\u53ef\u6c42\u5f97\u7ed3\u679c.\uff081\uff09\u5982\u56fe\uff1b \uff082\uff09\u2235AB\u662f\u2299O\u7684\u76f4\u5f84\uff0c\u2234\u2220ADB=90\u00b0\uff0c\u800cDE\u22a5AC\uff0c\u2234\u2220AED=90\u00b0\uff0c\u2235AD\u5e73\u5206\u2220CAB\uff0c\u2234\u2220CAD=\u2220DAB\uff0c\u2234Rt\u25b3ADE\u223dRt\u25b3ABD\uff0c\u2234AD\uff1aAB=AE\uff1aAD\uff0c\u2234AD 2 =AE?AB\uff1b\uff083\uff09\u8fdeOD\u3001BC\uff0c\u5b83\u4eec\u4ea4\u4e8e\u70b9G\uff0c\u5982\u56fe\uff0c \u22355AC=3AB\uff0c\u5373AC\uff1aAB=3\uff1a5\uff0c\u2234\u4e0d\u59a8\u8bbeAC=3x\uff0cAB=5x\uff0c\u2235AB\u662f\u2299O\u7684\u76f4\u5f84\uff0c\u2234\u2220ACB=90\u00b0\uff0c\u53c8\u2235\u2220CAD=\u2220DAB\uff0c\u2234\u5f27DC=\u5f27DB\uff0c\u2234OD\u5782\u76f4\u5e73\u5206BC\uff0c\u2234OD\u2225AE\uff0cOG= AC= \uff0c\u2234\u56db\u8fb9\u5f62ECGD\u4e3a\u77e9\u5f62\uff0c\u2234CE=DG=OD-OG= x- x =x\uff0c\u2234AE=AC+CE=3x+x=4x\uff0c\u2235AE\u2225OD\uff0c\u2234\u25b3AEF\u223d\u25b3DOF\uff0c\u2234AE\uff1aOD=EF\uff1aOF\uff0c\u2234EF\uff1aOF=4x\uff1a x=8\uff1a5\uff0c\u2234 \uff0e\u70b9\u8bc4\uff1a\u89e3\u7b54\u672c\u9898\u7684\u5173\u952e\u662f\u719f\u7ec3\u638c\u63e1\u76f4\u5f84\u6240\u5bf9\u7684\u5706\u5468\u89d2\u662f\u76f4\u89d2\uff1b\u76f8\u4f3c\u4e09\u89d2\u5f62\u7684\u5bf9\u5e94\u8fb9\u6210\u6bd4\u4f8b\uff0c\u6ce8\u610f\u5bf9\u5e94\u5b57\u6bcd\u5728\u5bf9\u5e94\u4f4d\u7f6e\u4e0a.
（1）解：如图；（2）证明：∵AB是⊙O的直径，
∴∠ADB=90°，
而DE⊥AC，
∴∠AED=90°，
∵AD平分∠CAB，
∴∠CAD=∠DAB，
∴Rt△ADE∽Rt△ABD，
∴AD：AB=AE：AD，
∴AD2=AE?AB；
（3）解：连OD、BC，它们交于点G，如图，
∵5AC=3AB，即AC：AB=3：5，
∴不妨设AC=3x，AB=5x，
∵AB是⊙O的直径，
∴∠ACB=90°，
又∵∠CAD=∠DAB，
∴弧DC=弧DB，
∴OD垂直平分BC，
∴OD∥AE，
∴OG=
| 1 |
| 2 |
| 3 |
| 2 |
∴四边形ECGD为矩形，
∴CE=DG=OD-OG=
| 5 |
| 2 |
| 3 |
| 2 |
∴AE=AC+CE=3x+x=4x，
∵AE∥OD，
∴△AEF∽△DOF，
∴AE：OD=EF：OF，
∴EF：OF=4x：
| 5 |
| 2 |
∴
| OE |
| OF |
| 8+5 |
| 5 |
| 13 |
| 5 |
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