document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206128>Question 1206128</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write a coordinate proof to show that the segments connecting the midpoints of any quadrilateral form a parallelogram. I think it is simpler to use vectors but not sure if I have to use midpoint formulas. Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843370 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=843370\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let the coordinates of the vertices of the quadrilateral be<br><BR>\n" );
document.write( "A(2a,2b)<BR>\n" );
document.write( "B(2c,2d)<BR>\n" );
document.write( "C(2e,2f)<BR>\n" );
document.write( "D(2g,2h)<br><BR>\n" );
document.write( "The midpoints are<br><BR>\n" );
document.write( "AB: P(a+c,b+d)<BR>\n" );
document.write( "BC: Q(c+e,d+f)<BR>\n" );
document.write( "CD: R(e+g,f+h)<BR>\n" );
document.write( "DA: S(g+a,h+b)<br><BR>\n" );
document.write( "Opposite sides PQ and RS have the same slope:<BR>\n" );
document.write( "PQ: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28b%2Bd%29-%28d%2Bf%29%29%2F%28%28a%2Bc%29-%28c%2Be%29%29=%28b-f%29%2F%28a-e%29 ALIGN=MIDDLE  ALT=\"%28%28b%2Bd%29-%28d%2Bf%29%29%2F%28%28a%2Bc%29-%28c%2Be%29%29=%28b-f%29%2F%28a-e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "RS: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28f%2Bh%29-%28h%2Bb%29%29%2F%28%28e%2Bg%29-%28g%2Ba%29%29=%28f-b%29%2F%28e-a%29=%28b-f%29%2F%28a-e%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And opposite side QR and PS have the same slope:<BR>\n" );
document.write( "QR: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28d%2Bf%29-%28f%2Bh%29%29%2F%28%28c%2Be%29-%28e%2Bg%29%29=%28d-h%29%2F%28c-g%29 ALIGN=MIDDLE  ALT=\"%28%28d%2Bf%29-%28f%2Bh%29%29%2F%28%28c%2Be%29-%28e%2Bg%29%29=%28d-h%29%2F%28c-g%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "PS: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28b%2Bd%29-%28h%2Bb%29%29%2F%28%28a%2Bc%29-%28g%2Ba%29%29=%28d-h%29%2F%28c-g%29 ALIGN=MIDDLE  ALT=\"%28%28b%2Bd%29-%28h%2Bb%29%29%2F%28%28a%2Bc%29-%28g%2Ba%29%29=%28d-h%29%2F%28c-g%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Both pairs of opposite sides are parallel, making the quadrilateral a parallelogram.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );