document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1120393>Question 1120393</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can there exist a triangle ROS in which the trisectors of angle O intersect RS at D and E with RD =1,,DE=2,andES=4? Explain \r<BR>\n" );
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document.write( "Thanks for any help .  </SPAN>\n" );
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document.write( "No; it is not possible.<br><BR>\n" );
document.write( "Consider first the fact that an angle bisector in a triangle divides the opposite side into two parts in the same ratio as the lengths of the two sides of the angle.<br><BR>\n" );
document.write( "In triangle ROE, OD is the angle bisector.  If RD=1 and DE=2, OE is twice the length of OR.  Let OR=x and OE=2x.<br><BR>\n" );
document.write( "In triangle DOS, OE is the angle bisector.  If DE=2 and ES=4, OS is twice the length of OD.  Let OD=y and OS=2y.<br><BR>\n" );
document.write( "Now use Stewart's Theorem in triangles ROE and DOS to get two equations relating x and y.<br><BR>\n" );
document.write( "Triangle ROE:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B4x%5E2+=+3y%5E2%2B6 ALIGN=MIDDLE  ALT=\"2x%5E2%2B4x%5E2+=+3y%5E2%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3y%5E2+=+6x%5E2-6 ALIGN=MIDDLE  ALT=\"3y%5E2+=+6x%5E2-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+=+2x%5E2-2 ALIGN=MIDDLE  ALT=\"y%5E2+=+2x%5E2-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Triangle DOS:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y%5E2%2B8y%5E2+=+24x%5E2%2B48 ALIGN=MIDDLE  ALT=\"4y%5E2%2B8y%5E2+=+24x%5E2%2B48\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12y%5E2+=+24x%5E2%2B48 ALIGN=MIDDLE  ALT=\"12y%5E2+=+24x%5E2%2B48\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+=+2x%5E2%2B4 ALIGN=MIDDLE  ALT=\"y%5E2+=+2x%5E2%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "But now we have two equations, based on the given information, that say in one case that y^2=2x^2-2 and in the other case y^2 = 2x^2+4.<br><BR>\n" );
document.write( "So the given conditions lead us to two equations that are incompatible; that means the given conditions are not possible. </SPAN>\n" );
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