document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1066085>Question 1066085</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If there are 7 distinct points on a plane with no three of which are collinear, how many different polygons can be possibly formed? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #681399 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=681399\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I would like the points to be around a circle and a requirement that the polygon be a convex polygon, and its vertices be some or all of the 7 points.<BR>\n" );
document.write( "I believe that would force the n-gon polygons to be set of n of the 7 points,<BR>\n" );
document.write( "instead of risking the answer being dependent on the relative position of the 7 points.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Otherwise, the relative positions of the points could be such that<BR>\n" );
document.write( "connecting several of points in random order<BR>\n" );
document.write( "could result in more or less closed polygonal lines that are not polygons.<BR>\n" );
document.write( "For example, with the points below<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.6%2C1.6%2C-1.6%2C1.6%2C%0D%0Acircle%28-0.5%2C-1%2C0.04%29%2Ccircle%280.5%2C-1%2C0.04%29%2C%0D%0Acircle%28-0.5%2C-0.3%2C0.04%29%2Ccircle%280.5%2C-0.3%2C0.04%29%2C%0D%0Acircle%28-1%2C0.3%2C0.04%29%2Ccircle%281%2C0.3%2C0.04%29%2C%0D%0Acircle%280%2C1%2C0.04%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the green polygonal line crosses itself, and cannot be considered a quadrilateral.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.6%2C1.6%2C-1.6%2C1.6%2C%0D%0Acircle%28-0.5%2C-1%2C0.04%29%2Ccircle%280.5%2C-1%2C0.04%29%2C%0D%0Acircle%28-0.5%2C-0.3%2C0.04%29%2Ccircle%280.5%2C-0.3%2C0.04%29%2C%0D%0Acircle%28-1%2C0.3%2C0.04%29%2Ccircle%281%2C0.3%2C0.04%29%2C%0D%0Acircle%280%2C1%2C0.04%29%2Cred%28rectangle%28-0.5%2C-1%2C0.5%2C-0.3%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.6%2C1.6%2C-1.6%2C1.6%2C%0D%0Acircle%28-0.5%2C-1%2C0.04%29%2Ccircle%280.5%2C-1%2C0.04%29%2C%0D%0Acircle%28-0.5%2C-0.3%2C0.04%29%2Ccircle%280.5%2C-0.3%2C0.04%29%2C%0D%0Acircle%28-1%2C0.3%2C0.04%29%2Ccircle%281%2C0.3%2C0.04%29%2C%0D%0Acircle%280%2C1%2C0.04%29%2Cgreen%28triangle%28-0.5%2C-1%2C-0.5%2C-0.3%2C0%2C-0.65%29%29%2C%0D%0Agreen%28triangle%280.5%2C-1%2C0.5%2C-0.3%2C0%2C-0.65%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I would say that the green closed polygonal is not a polygon.<BR>\n" );
document.write( "Should we consider it to be two triangles?<BR>\n" );
document.write( "What about the two polygonals below?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.6%2C1.6%2C-1.6%2C1.6%2C%0D%0Acircle%28-0.5%2C-1%2C0.04%29%2Ccircle%280.5%2C-1%2C0.04%29%2C%0D%0Acircle%28-0.5%2C-0.3%2C0.04%29%2Ccircle%280.5%2C-0.3%2C0.04%29%2C%0D%0Acircle%28-1%2C0.3%2C0.04%29%2Ccircle%281%2C0.3%2C0.04%29%2C%0D%0Acircle%280%2C1%2C0.04%29%2Cred%28line%28-0.5%2C-1%2C0.5%2C-1%29%29%2C%0D%0Ared%28line%28-0.5%2C-1%2C-0.5%2C-0.3%29%29%2Cred%28line%28-0.5%2C-0.3%2C-1%2C0.3%29%29%2C%0D%0Ared%28line%28-1%2C0.3%2C0%2C1%29%29%2Cred%28line%280%2C1%2C1%2C0.3%29%29%2C%0D%0Ared%28line%281%2C0.3%2C0.5%2C-0.3%29%29%2Cred%28line%280.5%2C-0.3%2C0.5%2C-1%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.6%2C1.6%2C-1.6%2C1.6%2C%0D%0Acircle%28-0.5%2C-1%2C0.04%29%2Ccircle%280.5%2C-1%2C0.04%29%2C%0D%0Acircle%28-0.5%2C-0.3%2C0.04%29%2Ccircle%280.5%2C-0.3%2C0.04%29%2C%0D%0Acircle%28-1%2C0.3%2C0.04%29%2Ccircle%281%2C0.3%2C0.04%29%2C%0D%0Acircle%280%2C1%2C0.04%29%2Cgreen%28line%28-0.5%2C-1%2C0.5%2C-1%29%29%2C%0D%0Agreen%28line%28-0.5%2C-1%2C-1%2C0.3%29%29%2Cgreen%28line%28-1%2C0.3%2C-0.5%2C-0.3%29%29%2C%0D%0Agreen%28line%280.5%2C-1%2C1%2C0.3%29%29%2Cgreen%28line%281%2C0.3%2C0.5%2C-0.3%29%29%2C%0D%0Agreen%28line%28-0.5%2C-0.3%2C0%2C1%29%29%2Cgreen%28line%280.5%2C-0.3%2C0%2C1%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > They are both non-convex heptagons.<BR>\n" );
document.write( "How many heptagons can you make with the 7 points below?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.1%2C1.1%2C-1.1%2C1.1%2C%0D%0Acircle%281%2C0%2C0.03%29%2Ccircle%280.623%2C0.782%2C0.03%29%2C%0D%0Acircle%28-0.223%2C0.975%2C0.03%29%2Ccircle%28-0.901%2C0.434%2C0.03%29%2C%0D%0Acircle%28-0.901%2C-0.434%2C0.03%29%2Ccircle%28-0.223%2C-0.975%2C0.03%29%2C%0D%0Acircle%280.623%2C-0.782%2C0.03%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );