document.write( "Question 1188287: 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? \n" ); document.write( "
Algebra.Com's Answer #848640 by yurtman(42)\"\" \"About 
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Certainly, let's find the ratio of BG:GF.\r
\n" ); document.write( "\n" ); document.write( "**1. Utilize Ceva's Theorem:**\r
\n" ); document.write( "\n" ); 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
\n" ); document.write( "\n" ); document.write( " (AF/FC) * (BD/DA) * (CE/EB) = 1\r
\n" ); document.write( "\n" ); document.write( "**2. Apply the given ratios:**\r
\n" ); document.write( "\n" ); document.write( "* BD/DA = 1/1 (since D is the midpoint of AB)
\n" ); document.write( "* BE/EC = 2/3 \r
\n" ); document.write( "\n" ); document.write( "**3. Calculate AF/FC:**\r
\n" ); document.write( "\n" ); document.write( "* Using Ceva's Theorem:
\n" ); document.write( " (AF/FC) * (1/1) * (3/2) = 1
\n" ); document.write( " AF/FC = 2/3\r
\n" ); document.write( "\n" ); document.write( "**4. Use Menelaus' Theorem:**\r
\n" ); document.write( "\n" ); 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
\n" ); document.write( "\n" ); document.write( " (AD/DB) * (BE/EC) * (CF/FA) = 1\r
\n" ); document.write( "\n" ); document.write( "**5. Apply the known ratios and solve for BG/GF:**\r
\n" ); document.write( "\n" ); document.write( "* (1/1) * (2/3) * (3/2) * (BG/GF) = 1
\n" ); document.write( " BG/GF = 1\r
\n" ); document.write( "\n" ); document.write( "**Therefore, the ratio of BG:GF is 1:1.**\r
\n" ); document.write( "\n" ); document.write( "Let me know if you have any other questions or problems to solve!
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