document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=692844>Question 692844</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A pleasant day Sir/ Madam, my daughter has this problem in geometry and I am not good in geometric proofs, can you please help me?\r<BR>\n" );
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document.write( "the illustration or drawing is a rectangle inscribed in a circle<BR>\n" );
document.write( "Given:  Quadrilateral ABCD is a rectangle<BR>\n" );
document.write( "Prove: line segment AC and line segment BD are diameters of the circle\r<BR>\n" );
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document.write( "Thank you very much for anyone who can help me! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #427124 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=427124\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "first draw the picture:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C600%2C+-5%2C+15%2C+-5%2C+10%2C+%0D%0A+++++++++grid%280%29%2Cblue%28+circle%28+5.5%2C+2.5%2C+4.9+%29+%29%2Clocate%28+5.5%2C+2.5%2C+O+%29%2Clocate%28+1%2C+1%2C+A+%29%2Clocate%2810%2C+1%2C+B%29%2Clocate%28+10%2C+4%2C+C+%29%2Clocate%28+1%2C+4%2C+D+%29%2C%0D%0A+++++++++line%28+1%2C+1%2C+10%2C+1+%29%2Cline%28+10%2C+1%2C+1%2C+4+%29%2Cline%28+1%2C+1%2C+10%2C+4+%29%2Cline%28+10%2C+1%2C+10%2C+4+%29%2Cline%281%2C+1%2C+1%2C+4+%29%2Cline%28+1%2C+4%2C+10%2C+4+%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "next, we will need to prove this:<BR>\n" );
document.write( "theorem: Diagonal of any rectangle inscribed in a circle is a diameter of the circle. (\r<BR>\n" );
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document.write( "This is essentially the converse of Thales' theorem.\r<BR>\n" );
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document.write( "Let A, B, C and D be the vertices of a rectangle inscribed in a circle and let AC and BD be the diagonals of this rectangle. \r<BR>\n" );
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document.write( "We can now focus on any one of the four triangles: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABC ALIGN=MIDDLE  ALT=\"ABC\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BCD ALIGN=MIDDLE  ALT=\"BCD\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CDA ALIGN=MIDDLE  ALT=\"CDA\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DAB ALIGN=MIDDLE  ALT=\"DAB\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's take <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABC ALIGN=MIDDLE  ALT=\"ABC\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC ALIGN=MIDDLE  ALT=\"AC\"   DISABLED_style= \"max-width:90%; height:auto;\" > side of this triangle is the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=diagonal ALIGN=MIDDLE  ALT=\"diagonal\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the rectangle\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB ALIGN=MIDDLE  ALT=\"AB\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC ALIGN=MIDDLE  ALT=\"BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > are two sides of the rectangle, which potentially may be of different length (but we will prove they must be the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=same ALIGN=MIDDLE  ALT=\"same\"   DISABLED_style= \"max-width:90%; height:auto;\" > if the area of the inscribed rectangle is maximized).<BR>\n" );
document.write( "The angle at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the right angle since it is one of the angles of the rectangle, which by definition has four right angles.\r<BR>\n" );
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document.write( "Hence, looking at the triangle<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ABC ALIGN=MIDDLE  ALT=\"+ABC\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can see that this is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=right-angled ALIGN=MIDDLE  ALT=\"right-angled\"   DISABLED_style= \"max-width:90%; height:auto;\" > triangle inscribed in the circle.\r<BR>\n" );
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document.write( "Therefore to prove (1), we need to show that side<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+AC ALIGN=MIDDLE  ALT=\"+AC\"   DISABLED_style= \"max-width:90%; height:auto;\" > of any such triangle must be a  diameter of circle.\r<BR>\n" );
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document.write( "Proof: \r<BR>\n" );
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document.write( "Choose any three points <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the circle and connect these points to make a triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABC ALIGN=MIDDLE  ALT=\"ABC\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Let's suppose that the claim is that the angle at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > is right angle.\r<BR>\n" );
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document.write( "We will show that if this is true then it must follow that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC ALIGN=MIDDLE  ALT=\"AC\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=diameter ALIGN=MIDDLE  ALT=\"diameter\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=circle ALIGN=MIDDLE  ALT=\"circle\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Connect the center of the circle (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O ALIGN=MIDDLE  ALT=\"O\"   DISABLED_style= \"max-width:90%; height:auto;\" >) with each of the vertices of the triangle creating the segments <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OA ALIGN=MIDDLE  ALT=\"OA\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OB ALIGN=MIDDLE  ALT=\"OB\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OC ALIGN=MIDDLE  ALT=\"OC\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Let's call the angle defined by the path <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OAB ALIGN=MIDDLE  ALT=\"OAB\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the angle defined by the path <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OCB ALIGN=MIDDLE  ALT=\"OCB\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta ALIGN=MIDDLE  ALT=\"beta\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OA ALIGN=MIDDLE  ALT=\"OA\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OC+ ALIGN=MIDDLE  ALT=\"OC+\"   DISABLED_style= \"max-width:90%; height:auto;\" >and<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+OB ALIGN=MIDDLE  ALT=\"+OB\"   DISABLED_style= \"max-width:90%; height:auto;\" > are all of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=equal ALIGN=MIDDLE  ALT=\"equal\"   DISABLED_style= \"max-width:90%; height:auto;\" > length (equal to the length of the  radius  of the circle ) then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=all ALIGN=MIDDLE  ALT=\"all\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+three ALIGN=MIDDLE  ALT=\"+three\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+inner ALIGN=MIDDLE  ALT=\"+inner\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+triangles ALIGN=MIDDLE  ALT=\"+triangles\"   DISABLED_style= \"max-width:90%; height:auto;\" > (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OAB ALIGN=MIDDLE  ALT=\"OAB\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OAC ALIGN=MIDDLE  ALT=\"OAC\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OBC ALIGN=MIDDLE  ALT=\"OBC\"   DISABLED_style= \"max-width:90%; height:auto;\" >) are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=isosceles ALIGN=MIDDLE  ALT=\"isosceles\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "We will next want to find the angle between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OA ALIGN=MIDDLE  ALT=\"OA\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OC+ ALIGN=MIDDLE  ALT=\"OC+\"   DISABLED_style= \"max-width:90%; height:auto;\" >(the angle at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O ALIGN=MIDDLE  ALT=\"O\"   DISABLED_style= \"max-width:90%; height:auto;\" > made out by segments <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OA ALIGN=MIDDLE  ALT=\"OA\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OC ALIGN=MIDDLE  ALT=\"OC\"   DISABLED_style= \"max-width:90%; height:auto;\" >) using only angles <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=beta ALIGN=MIDDLE  ALT=\"beta\"   DISABLED_style= \"max-width:90%; height:auto;\" >as given.<BR>\n" );
document.write( "The angle between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OA ALIGN=MIDDLE  ALT=\"OA\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OB ALIGN=MIDDLE  ALT=\"OB\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180+-+2alpha ALIGN=MIDDLE  ALT=\"180+-+2alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" >° (due to the fact that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OAB ALIGN=MIDDLE  ALT=\"OAB\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=isosceles ALIGN=MIDDLE  ALT=\"isosceles\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the fact that the sum of all three internal angles in a triangle sum to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180 ALIGN=MIDDLE  ALT=\"180\"   DISABLED_style= \"max-width:90%; height:auto;\" >°).\r<BR>\n" );
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document.write( "Similarly, the angle between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OB ALIGN=MIDDLE  ALT=\"OB\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OC ALIGN=MIDDLE  ALT=\"OC\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equal to  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180+-+2beta ALIGN=MIDDLE  ALT=\"180+-+2beta\"   DISABLED_style= \"max-width:90%; height:auto;\" >°.\r<BR>\n" );
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document.write( "Finally, since all the three angles at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O ALIGN=MIDDLE  ALT=\"O\"   DISABLED_style= \"max-width:90%; height:auto;\" > add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360 ALIGN=MIDDLE  ALT=\"360\"   DISABLED_style= \"max-width:90%; height:auto;\" >° (full circle), it follows that the angle between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OA ALIGN=MIDDLE  ALT=\"OA\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OC ALIGN=MIDDLE  ALT=\"OC\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360+-+%28180+-+alpha%29+%96+%28180+-+beta%29+=+2%28alpha%2Bbeta%29 ALIGN=MIDDLE  ALT=\"360+-+%28180+-+alpha%29+%96+%28180+-+beta%29+=+2%28alpha%2Bbeta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "However, the angle at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the original triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABC ALIGN=MIDDLE  ALT=\"ABC\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28alpha%2Bbeta%29 ALIGN=MIDDLE  ALT=\"%28alpha%2Bbeta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (follows from the fact that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OAB ALIGN=MIDDLE  ALT=\"OAB\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OBC ALIGN=MIDDLE  ALT=\"OBC\"   DISABLED_style= \"max-width:90%; height:auto;\" > triangles are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=isosceles ALIGN=MIDDLE  ALT=\"isosceles\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "This angle was claimed to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90 ALIGN=MIDDLE  ALT=\"90\"   DISABLED_style= \"max-width:90%; height:auto;\" >° and therefore the angle between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OA ALIGN=MIDDLE  ALT=\"OA\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OC ALIGN=MIDDLE  ALT=\"OC\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=180 ALIGN=MIDDLE  ALT=\"180\"   DISABLED_style= \"max-width:90%; height:auto;\" >°.\r<BR>\n" );
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document.write( "This means that the points <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=center ALIGN=MIDDLE  ALT=\"center\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the circle (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O ALIGN=MIDDLE  ALT=\"O\"   DISABLED_style= \"max-width:90%; height:auto;\" >) are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=collinear ALIGN=MIDDLE  ALT=\"collinear\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "In other words, the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=center ALIGN=MIDDLE  ALT=\"center\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the circle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lying ALIGN=MIDDLE  ALT=\"lying\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the straight line segment <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC ALIGN=MIDDLE  ALT=\"AC\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which in turn means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AC ALIGN=MIDDLE  ALT=\"AC\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+must ALIGN=MIDDLE  ALT=\"+must\"   DISABLED_style= \"max-width:90%; height:auto;\" > be a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=diameter+ ALIGN=MIDDLE  ALT=\"diameter+\"   DISABLED_style= \"max-width:90%; height:auto;\" >of the<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+circle ALIGN=MIDDLE  ALT=\"+circle\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So, we can conclude that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=each ALIGN=MIDDLE  ALT=\"each\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+diagonal ALIGN=MIDDLE  ALT=\"+diagonal\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the rectangle inscribed in a circle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=must ALIGN=MIDDLE  ALT=\"must\"   DISABLED_style= \"max-width:90%; height:auto;\" > be a circles <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=diameter ALIGN=MIDDLE  ALT=\"diameter\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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