【竞赛】P是⊙O外一点PA切⊙O于A，PBC是⊙O的割线AD⊥PO于D，求证：PB:BD=PC:CD

<p><span class="Apple-style-span" style="widows: 2; text-transform: none; text-indent: 0px; border-collapse: separate; font: medium Simsun; white-space: normal; orphans: 2; letter-spacing: normal; color: rgb(0,0,0); word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px"><span class="Apple-style-span" style="line-height: 22px; font-family: Arial; font-size: 14px; font-weight: bold">题目：P是⊙O外一点，PA切⊙O于A，PBC是⊙O的割线，AD&perp;PO于D，求证：PB:BD=PC:CD&nbsp; （<span lang="EN-US" style="font-family: 新宋体; mso-bidi-font-size: 10.5pt">2002</span><span style="font-family: 新宋体; mso-bidi-font-size: 10.5pt">年四川省初中数学竞赛第四题</span><span lang="EN-US" style="font-size: 22pt; mso-bidi-font-size: 12.0pt"><o:p></o:p></span>）</span></span></p> <p><span><br /> 思路：<span style="mso-spacerun: yes"><font face="Times New Roman"> </font></span></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">因所证比例线段不是对应边，故不能通过判定△</span><span lang="EN-US"><font face="Times New Roman">PBD</font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">与△</span><span lang="EN-US"><font face="Times New Roman">PCD</font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">相似证明．连结</span><span lang="EN-US"><font face="Times New Roman">OA </font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">、</span><span lang="EN-US"><font face="Times New Roman">OB</font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">、</span><span lang="EN-US"><font face="Times New Roman"> OC</font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">，</span><font face="Times New Roman"> </font><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">由射影定理得</span><span lang="EN-US"><font face="Times New Roman">PA²=PD</font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">&middot;</span><span lang="EN-US"><font face="Times New Roman">PO </font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">由切割线定理得</span><span lang="EN-US"><font face="Times New Roman"> PA²=PB</font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">&middot;</span><span lang="EN-US"><font face="Times New Roman">PC</font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">，所以</span><span lang="EN-US"><font face="Times New Roman">PD</font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">&middot;</span><span lang="EN-US"><font face="Times New Roman">PO=PB</font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">&middot;</span><span lang="EN-US"><font face="Times New Roman">PC </font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">所以</span><span lang="EN-US"><font face="Times New Roman">B</font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">、</span><span lang="EN-US"><font face="Times New Roman">C</font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">、</span><span lang="EN-US"><font face="Times New Roman">O</font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">、</span><span lang="EN-US"><font face="Times New Roman">D</font></span><span style="font-family: 宋体; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">共圆，就得多对相似三角形，以此达到证明的目的．</span><span lang="EN-US"><o:p><font face="Times New Roman">&nbsp;<br /> </font></o:p></span></p> <p class="MsoNormal" style="text-indent: 10pt; margin: 0cm 0cm 0pt; mso-char-indent-count: 1.0"><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"><v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"></v:path><o:lock v:ext="edit" aspectratio="t"></o:lock></v:shapetype><v:shape id="_x0000_s1026" type="#_x0000_t75" style="z-index: 1; position: absolute; text-align: left; margin-top: 7.8pt; width: 83.55pt; height: 101.4pt; margin-left: 324pt; left: 0px; mso-position-horizontal: absolute; mso-position-vertical: absolute"><v:imagedata src="file:///C:\DOCUME~1\ADMINI~1\LOCALS~1\Temp\msohtml1\01\clip_image001.png" o:title=""></v:imagedata></v:shape><span style="font-family: Arial"><span style="mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">△</span><span lang="EN-US">PCD~</span><span style="mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">△</span><span lang="EN-US">POB </span><span style="mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">则</span> PC:CD=PO:OB</span></p> <p class="MsoNormal" style="text-indent: 10.5pt; margin: 0cm 0cm 0pt; mso-char-indent-count: 1.0"><span style="font-family: Arial"><span style="mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">△</span><span lang="EN-US">PBD~</span><span style="mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">△</span><span lang="EN-US">POC </span><span style="mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">则</span>&nbsp; PB:BD=PO:OC</span></p> <p class="MsoNormal" style="text-indent: 10.5pt; margin: 0cm 0cm 0pt; mso-char-indent-count: 1.0"><span style="font-family: Arial"><span style="mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">因为OB=OC</span><span lang="EN-US"><span style="position: relative; top: 3pt; mso-text-raise: -3.0pt">&nbsp;&nbsp;</span></span><span style="mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'">所以 PB:BD=PC:CD</span></span></p> <p>&nbsp;</p> <p>&nbsp;</p> <p>&nbsp;<img alt="" width="305" height="339" src="/download.asp?id=42" /><br /> &nbsp;<br /> 本题的题目和解答过程WORD格式文件下载&nbsp; <a href="/download.asp?id=43"><img border="0" alt="" style="margin: 0px 2px -4px 0px" src="/images/download.gif" />点击下载此文件</a></p>

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