document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178381>Question 1178381</A>: <I><SPAN STYLE=\"word-break: break-all;\"> five points O,A,B,C,D are taken in order on a straight line with distances OA=a, OB=b,OC=c,and OD=d. <BR>\n" );
document.write( "P is a point on the line between B and C and such that AP:PD=BP:PC.<BR>\n" );
document.write( "Find OP in terms of a,b,c,d. </SPAN>\n" );
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document.write( "Let BP=x.  Then<br><BR>\n" );
document.write( "AP = b+x<BR>\n" );
document.write( "PD = d-(b+x)<BR>\n" );
document.write( "BP = x<BR>\n" );
document.write( "PC = c-(b+x)<br><BR>\n" );
document.write( "AP/PD = (b+x)/(d-(b+x))<br><BR>\n" );
document.write( "BP/PC = x/(c-(b+x))<br><BR>\n" );
document.write( "AP/PD = BP/PC:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2Bx%29%2F%28d-%28b%2Bx%29%29+=+x%2F%28c-%28b%2Bx%29%29 ALIGN=MIDDLE  ALT=\"%28b%2Bx%29%2F%28d-%28b%2Bx%29%29+=+x%2F%28c-%28b%2Bx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%28b%2Bx%29-%28b%2Bx%29%5E2+=+dx-bx-x%5E2 ALIGN=MIDDLE  ALT=\"c%28b%2Bx%29-%28b%2Bx%29%5E2+=+dx-bx-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cb%2Bcx-b%5E2-2bx-x%5E2+=+dx-bx-x%5E2 ALIGN=MIDDLE  ALT=\"cb%2Bcx-b%5E2-2bx-x%5E2+=+dx-bx-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cb%2Bcx-b%5E2-2bx+=+dx-bx ALIGN=MIDDLE  ALT=\"cb%2Bcx-b%5E2-2bx+=+dx-bx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cb-b%5E2+=+dx-bx-cx%2B2bx ALIGN=MIDDLE  ALT=\"cb-b%5E2+=+dx-bx-cx%2B2bx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cb-b%5E2+=+bx%2Bdx-cx ALIGN=MIDDLE  ALT=\"cb-b%5E2+=+bx%2Bdx-cx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cb-b%5E2+=+x%28b%2Bd-c%29 ALIGN=MIDDLE  ALT=\"cb-b%5E2+=+x%28b%2Bd-c%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28cb-b%5E2%29%2F%28b%2Bd-c%29 ALIGN=MIDDLE  ALT=\"x+=+%28cb-b%5E2%29%2F%28b%2Bd-c%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OP+=+b%2Bx+=+b%2B%28cb-b%5E2%29%2F%28b%2Bd-c%29+=+%28%28b%5E2%2Bbd-bc%29%2B%28cb-b%5E2%29%29%2F%28b%2Bd-c%29+=+bd%2F%28b%2Bd-c%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OP+=+bd%2F%28b%2Bd-c%29 ALIGN=MIDDLE  ALT=\"OP+=+bd%2F%28b%2Bd-c%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );