document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1188287>Question 1188287</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given triangle ABC with D the midpoint of side AB and E on side BC, F on side AC and G is the intersection of side DE and side BF. BE:EC = 2:3 and DG:GE = 5:8. What is the ratio of BG:GF? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848640 by </B><A HREF=/tutors/aboutme.mpl?userid=yurtman><B>yurtman(42)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=yurtman><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848640\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's find the ratio of BG:GF.\r<BR>\n" );
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document.write( "**1. Utilize Ceva's Theorem:**\r<BR>\n" );
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document.write( "* Ceva's Theorem states that for any triangle ABC, if lines AD, BE, and CF intersect at a single point (in this case, point G), then:\r<BR>\n" );
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document.write( "   (AF/FC) * (BD/DA) * (CE/EB) = 1\r<BR>\n" );
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document.write( "**2. Apply the given ratios:**\r<BR>\n" );
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document.write( "* BD/DA = 1/1 (since D is the midpoint of AB)<BR>\n" );
document.write( "* BE/EC = 2/3 \r<BR>\n" );
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document.write( "**3. Calculate AF/FC:**\r<BR>\n" );
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document.write( "* Using Ceva's Theorem:<BR>\n" );
document.write( "   (AF/FC) * (1/1) * (3/2) = 1<BR>\n" );
document.write( "   AF/FC = 2/3\r<BR>\n" );
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document.write( "**4. Use Menelaus' Theorem:**\r<BR>\n" );
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document.write( "* Menelaus' Theorem states that for any transversal line (in this case, line DE) that intersects the sides of a triangle (triangle ABC) at points D, E, and F, then:\r<BR>\n" );
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document.write( "   (AD/DB) * (BE/EC) * (CF/FA) = 1\r<BR>\n" );
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document.write( "**5. Apply the known ratios and solve for BG/GF:**\r<BR>\n" );
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document.write( "* (1/1) * (2/3) * (3/2) * (BG/GF) = 1 <BR>\n" );
document.write( "   BG/GF = 1\r<BR>\n" );
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document.write( "**Therefore, the ratio of BG:GF is 1:1.**\r<BR>\n" );
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document.write( "Let me know if you have any other questions or problems to solve! <BR>\n" );
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