document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166456>Question 1166456</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The complex numbers z1, z2 and z3 are represented in the complex plane by the points P, Q and R respectively. If the line segments PQ and PR have the same length and are perpendicular to one another, prove:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28z1%29%5E2+%2B+%28z2%29%5E2+%2B+%28z3%29%5E2 ALIGN=MIDDLE  ALT=\"2%28z1%29%5E2+%2B+%28z2%29%5E2+%2B+%28z3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2z1%28z2%2Bz3%29 ALIGN=MIDDLE  ALT=\"2z1%28z2%2Bz3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I've tried multiplying the modulus of PQ and PR together, and the answer I get is close but not quite right.\r<BR>\n" );
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document.write( "Help? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #791003 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=791003\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Assume P,Q,and R are oriented in an arbitrary way WRT a set of coordinate axes (but the points obey the constraints given by the problem statement).<BR>\n" );
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document.write( "Now rotate and translate a set of coordinate axes (WLOG) such that Q is at the origin <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2B0i ALIGN=MIDDLE  ALT=\"0%2B0i\"   DISABLED_style= \"max-width:90%; height:auto;\" >, P is at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D%2Bb%5B1%5Di ALIGN=MIDDLE  ALT=\"a%5B1%5D%2Bb%5B1%5Di\"   DISABLED_style= \"max-width:90%; height:auto;\" > and R is at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D+%2B+0i ALIGN=MIDDLE  ALT=\"a%5B3%5D+%2B+0i\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C+-2%2C+8%2C+-2%2C+8%2C+grid%280%29%2C%0D%0A+++++++++++line%280%2C0%2C3%2C3%29%2C%0D%0A+++++++++++line%283%2C3%2C6%2C0%29%2C%0D%0A+++++++++++locate%280.1%2C-0.2%2CQ%29%2C+%0D%0A+++++++++++locate%283%2C3.8%2CP+%28a%5B1%5D%2Bb%5B1%5Di%29%29%2C%0D%0A+++++++++++locate%286.1%2C1.1%2CR+%28a%5B3%5D%29%29%0D%0A%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then you get:<BR>\n" );
document.write( "LHS = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2z%5B1%5D%5E2%2Bz%5B2%5D%5E2%2Bz%5B3%5D%5E2+=+2z%5B1%5D%5E2+%2B+a%5B3%5D%5E2+ ALIGN=MIDDLE  ALT=\"2z%5B1%5D%5E2%2Bz%5B2%5D%5E2%2Bz%5B3%5D%5E2+=+2z%5B1%5D%5E2+%2B+a%5B3%5D%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "RHS = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2z%5B1%5D%28z%5B2%5D%2Bz%5B3%5D%29+=+2%28a%5B1%5D%2Bb%5B1%5Di%29%28a%5B3%5D%29+=+2a%5B1%5Da%5B3%5D%2B2a%5B3%5Db%5B1%5Di+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Apply Pythagorean Theorem <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+abs%28PQ%29%5E2+%2B+abs%28PR%29%5E2+=+abs%28QR%29%5E2+ ALIGN=MIDDLE  ALT=\"+abs%28PQ%29%5E2+%2B+abs%28PR%29%5E2+=+abs%28QR%29%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=+%28a%5B1%5D-a%5B3%5D%29%5E2%2Bb%5B1%5D%5E2%2Ba%5B1%5D%5E2%2Bb%5B1%5D%5E2+=+a%5B3%5D%5E2+ ALIGN=MIDDLE  ALT=\"+%28a%5B1%5D-a%5B3%5D%29%5E2%2Bb%5B1%5D%5E2%2Ba%5B1%5D%5E2%2Bb%5B1%5D%5E2+=+a%5B3%5D%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "...<BR>\n" );
document.write( "   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28a%5B1%5D%5E2+%2B+b%5B1%5D%5E2%29+=+2a%5B1%5Da%5B3%5D+ ALIGN=MIDDLE  ALT=\"2%28a%5B1%5D%5E2+%2B+b%5B1%5D%5E2%29+=+2a%5B1%5Da%5B3%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Note that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D+=+2a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B3%5D+=+2a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > so the LHS becomes:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2z%5B1%5D%5E2+%2B+a%5B3%5D%5E2+=+2z%5B1%5D%5E2+%2B+4a%5B1%5D%5E2+ ALIGN=MIDDLE  ALT=\"2z%5B1%5D%5E2+%2B+a%5B3%5D%5E2+=+2z%5B1%5D%5E2+%2B+4a%5B1%5D%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <br><BR>\n" );
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document.write( "and using the Pythagorean result, the RHS becomes:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2a%5B1%5Da%5B3%5D%2B2a%5B3%5Db%5B1%5Di+=+2a%5B1%5D%5E2%2B2b%5B1%5D%5E2+%2B+4a%5B1%5Db%5B1%5Di+=+2z%5B1%5D%5E2+%2B+4b%5B1%5D%5E2+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "Drop a perpendicular from P to the x-axis, the height of P from the x-axis and horizontal distance from origin are the same, these are also <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B1%5D+ ALIGN=MIDDLE  ALT=\"b%5B1%5D+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > respectively.  Thus, we have shown <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D+=+b%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D+=+b%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "Siince <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D+=+b%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D+=+b%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > the proof is complete.\r<BR>\n" );
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document.write( "Undoing the translation and rotation of the coordinate axes back to whatever the 'original' position was changes nothing (but the algebra is more complicated in that orientation).\r<BR>\n" );
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document.write( " </SPAN>\n" );
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