document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=843884>Question 843884</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a polygon has n sides, how many triangles are formed by drawing all diagonals from one vertex?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #508409 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=508409\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I would make that polygon a convex polygon,<BR>\n" );
document.write( "like this: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-5%2C5%2C-1%2C9%2C%0D%0Aline%28-2%2C0%2C3%2C0%29%2Cline%28-2%2C0%2C-4%2C4%29%2C%0D%0Aline%28-4%2C4%2C-4%2C7%29%2Cline%28-4%2C7%2C-1%2C9%29%2C%0D%0Aline%28-1%2C9%2C1%2C8%29%2Cline%281%2C8%2C3%2C6%29%2C%0D%0Aline%283%2C6%2C5%2C2%29%2Cline%283%2C0%2C5%2C2%29%2C%0D%0Agreen%28line%283%2C0%2C3%2C6%29%29%2Cgreen%28line%283%2C0%2C1%2C8%29%29%2C%0D%0Agreen%28line%283%2C0%2C-1%2C9%29%29%2Cgreen%28line%283%2C0%2C-4%2C7%29%29%2C%0D%0Agreen%28line%283%2C0%2C-4%2C4%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > so all diagonals will be inside the polygon, and no triangles will overlap.<BR>\n" );
document.write( "The answer may be the same for a concave polygon, like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-5%2C5%2C-1%2C9%2C%0D%0Aline%28-2%2C0%2C3%2C0%29%2Cline%28-2%2C0%2C-1%2C4%29%2C%0D%0Aline%28-1%2C4%2C-4%2C3%29%2Cline%28-4%2C3%2C2%2C7%29%2C%0D%0Aline%282%2C7%2C-1%2C8%29%2Cline%28-1%2C8%2C3%2C9%29%2C%0D%0Aline%283%2C9%2C5%2C2%29%2Cline%283%2C0%2C5%2C2%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , but it is harder to visualize with concave polygons.<BR>\n" );
document.write( "I believe the answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28n-2%29 ALIGN=MIDDLE  ALT=\"highlight%28n-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A polygon with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > sides has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > vertices.<BR>\n" );
document.write( "From one of the n vertices, you can draw <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-3 ALIGN=MIDDLE  ALT=\"n-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > diagonals to all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-1 ALIGN=MIDDLE  ALT=\"n-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > other vertices except the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > adjacent ones.<BR>\n" );
document.write( "(The lines connecting to the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > adjacent vertices are sides of the polygon, and we cannot call them diagonals).<BR>\n" );
document.write( "To one side of each diagonal is a triangle, and you count <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-3 ALIGN=MIDDLE  ALT=\"n-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > of those:<BR>\n" );
document.write( "one to that side of the first diagonal,<BR>\n" );
document.write( "a second one to that side of the second diagonal, and so on.<BR>\n" );
document.write( "You count <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-3 ALIGN=MIDDLE  ALT=\"n-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > triangles that way.<BR>\n" );
document.write( "There is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > more triangle to the other side of the last of those diagonals. </SPAN>\n" );
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