document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=513348>Question 513348</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Proof that the diagonals of a parallelogram bisect eachother </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #342963 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=342963\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<b>Theorem</b> If ABCD is a <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Rectangles/Different-between-parallelogram-rectangle-square-rhombus-and-trapezoid.lesson>parallelogram</A>, then prove that the diagonals of ABCD bisect each other.\r<BR>\n" );
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document.write( "<b>Proof</b>\r<BR>\n" );
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document.write( "Let the two diagonals be AC and BD and O be the intersection point.\r<BR>\n" );
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document.write( "We have to prove that O is the midpoint of AC and also the midpoint of BD.\r<BR>\n" );
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document.write( "Hence, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AO=OC ALIGN=MIDDLE  ALT=\"AO=OC\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BO=OD ALIGN=MIDDLE  ALT=\"BO=OD\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We will prove using <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Triangles/Geometry-Proving-Triangles-Congruent.lesson>congruent triangles </A>concept.\r<BR>\n" );
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document.write( "Consider two <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Triangles/Triangles-and-its-basic-properties.lesson>Triangles </A> ABO and COD.\r<BR>\n" );
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document.write( "1. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+angle+OAB+=+angle+CAB+=+angle+ACD+=+angle+OCD ALIGN=MIDDLE  ALT=\"+angle+OAB+=+angle+CAB+=+angle+ACD+=+angle+OCD\"   DISABLED_style= \"max-width:90%; height:auto;\" >   ....( Line AC is a transversal of the parallel lines AB and CD, hence alternate angles).\r<BR>\n" );
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document.write( "2. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=angle+ODC+=+angle+BDC+=+angle+DBA+=+angle+OBA ALIGN=MIDDLE  ALT=\"angle+ODC+=+angle+BDC+=+angle+DBA+=+angle+OBA\"   DISABLED_style= \"max-width:90%; height:auto;\" >   ....(Line BD is a transversal of the parallel lines AB and CD, hence alternate angles).\r<BR>\n" );
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document.write( "3. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=angle+DOC+=+angle+AOB ALIGN=MIDDLE  ALT=\"angle+DOC+=+angle+AOB\"   DISABLED_style= \"max-width:90%; height:auto;\" >                           ....(Opposite angles when two lines intersect each other area equal)\r<BR>\n" );
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document.write( "From conditions 1,2 and 3\r<BR>\n" );
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document.write( "Triangle ABO is similar to triangle CDO (By <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Triangles/Angle.wikipedia>Angle </A>-Angle similar property)\r<BR>\n" );
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document.write( "Since Triangles are similar, Hence ratio of sides are equal from <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Triangles/Geometry-Similar-Triangles.lesson>similar triangles </A>property.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28DC%2FAB%29=%28DO%2FOB%29=%28CO%2FOA%29 ALIGN=MIDDLE  ALT=\"%28DC%2FAB%29=%28DO%2FOB%29=%28CO%2FOA%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >     .........(4)\r<BR>\n" );
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document.write( "From theorem that <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Parallelograms/Opposite-sides-of-a-parallelogram-are-equal.lesson>Opposite sides of a parallelogram are equal</A>,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DC=AB ALIGN=MIDDLE  ALT=\"DC=AB\"   DISABLED_style= \"max-width:90%; height:auto;\" >                       ..........(5)\r<BR>\n" );
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document.write( "From equation (4) and (5)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28DC%2FAB%29=%28DO%2FOB%29=%28CO%2FOA%29=1 ALIGN=MIDDLE  ALT=\"%28DC%2FAB%29=%28DO%2FOB%29=%28CO%2FOA%29=1\"   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=DO%2FOB+=+1 ALIGN=MIDDLE  ALT=\"DO%2FOB+=+1\"   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=DO+=+OB ALIGN=MIDDLE  ALT=\"DO+=+OB\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Similarly, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CO=OA ALIGN=MIDDLE  ALT=\"CO=OA\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Hence, We conclude that AO = CO and BO = DO.\r<BR>\n" );
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document.write( "QED\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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