document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183719>Question 1183719</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given regular hexagon ABCDEF, with center O and sides of length a.  Let G be the midpoint of BC.  Let H be the midpoint of DE.  AH intersects EB at J and FG intersect EB at K.  Find JK.  (Hint:  Draw auxiliary lines HG and DA. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814165 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</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=814165\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "I haven't come up with a standard geometric solution; but a solution using coordinate geometry works nicely.<br><BR>\n" );
document.write( "Let the origin (0,0) be the center of the regular hexagon; and for simplicity choose a \"nice\" number for the side length -- I chose 12.  My hexagon then has vertices<BR>\n" );
document.write( "A(-12,0)<BR>\n" );
document.write( "B(-6,6sqrt(3))<BR>\n" );
document.write( "C(6,6sqrt(3))<BR>\n" );
document.write( "D(12,0)<BR>\n" );
document.write( "E(6,-6sqrt(3))<BR>\n" );
document.write( "F(-6,6sqrt(3))<br><BR>\n" );
document.write( "From those, we get<BR>\n" );
document.write( "G(0,6sqrt(3)) [midpoint of BC]<BR>\n" );
document.write( "H(9,-3sqrt(3)) [midpoint of DE]<br><BR>\n" );
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document.write( "Find the equations of lines AH, BE, and FG<br><BR>\n" );
document.write( "BE: y=(-sqrt(3))x<BR>\n" );
document.write( "FG: y=(2sqrt(3))x+6sqrt(3)<BR>\n" );
document.write( "AH: slope = (-3sqrt(3)/21) = -sqrt(3)/7<BR>\n" );
document.write( " equation: y=(-sqrt(3)/7)(x+12) = (-sqrt(3)/7)x-12sqrt(3)/7<br><BR>\n" );
document.write( "Find the coordinates of K, the intersection of FG and EB<br><BR>\n" );
document.write( "(-sqrt(3)x)=2sqrt(3)x+6sqrt(3)<BR>\n" );
document.write( "-3sqrt(3)x=6sqrt(3)<BR>\n" );
document.write( "x=-2<BR>\n" );
document.write( "y=-sqrt(3)x = 2sqrt(3)<br><BR>\n" );
document.write( "Find the coordinates of J, the intersection of AH and BE<br><BR>\n" );
document.write( "(-sqrt(3)x)=(-sqrt(3)/7)x-12sqrt(3)/7<BR>\n" );
document.write( "(-7sqrt(3)x=-sqrt(3)x-12sqrt(3)<BR>\n" );
document.write( "-6sqrt(3)x=-12sqrt(3)<BR>\n" );
document.write( "x=2<BR>\n" );
document.write( "y=-sqrt(3)x=-2sqrt(3)<br><BR>\n" );
document.write( "So J is (2,-2sqrt(3)) and K is (-2,2sqrt(3))<br><BR>\n" );
document.write( "The length of JK is then given by the distance formula<BR>\n" );
document.write( "(JK)=sqrt(4^2+(4sqrt(3))^2) = sqrt(16+48) = sqrt(64) = 8<br><BR>\n" );
document.write( "ANSWER: JK = 8<br><BR>\n" );
document.write( "Or, since I used a=12,<br><BR>\n" );
document.write( "ANSWER: JK = (2/3)a<br><BR>\n" );
document.write( "Note the symmetry of the two points J and K with respect to the center of the hexagon suggests that a standard geometric solution to the problem should be possible; but I haven't yet seen it.<br><BR>\n" );
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