document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1187405>Question 1187405</A>: <I><SPAN STYLE=\"word-break: break-all;\"> https://imgur.com/HQvOuKW In the triangle, each vertex is joined to four points on the opposite side of the triangle, with no three lines intersecting at one point. How many non-overlapping regions are formed in the triangle? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #818424 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=818424\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "There might be some fancy mathematical formula for finding the answer to a problem like this, but it's probably more complicated than finding the answer by just counting.<br><BR>\n" );
document.write( "What is required is a systematic method for counting the regions so that each region gets counted exactly once.  Here is what I suggest.<br><BR>\n" );
document.write( "(1) Pick one vertex of the big triangle and consider the 6 lines from that vertex to the opposite side (2 lines to the other two vertices, and 4 lines to the opposite side).  Those lines divide the big triangle into 5 triangular regions.<BR>\n" );
document.write( "(2) Count (carefully, and perhaps at least twice) the number of regions in each of those 5 triangular regions.  As a check on your counting, note that, by symmetry, the numbers of regions in the 1st and 5th triangular regions should be the same, as should the numbers of regions in the 2nd and 4th.<br><BR>\n" );
document.write( "Happy counting!<br><br><BR>\n" );
document.write( "NOTE: My count was 61<br><BR>\n" );
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