document.write( "Question 1054195: ABCD is a parallelogram, in which E is the midpoint of AD and O is a point on AC such that AO=1/4AC.When EO produced meets AB at F.prove that F is mid point of AB \n" ); document.write( "

Algebra.Com's Answer #669563 by ikleyn(52782) $\"\"$ $\"About$

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\n" ); document.write( "ABCD is a parallelogram, in which E is the midpoint of AD and O is a point on AC such that AO=1/4AC.
\n" ); document.write( "When EO is produced it meets AB at F. Prove that F is mid point of AB
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document.write( "Make a sketch. \r\n" );
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document.write( "I have my sketch before my eyes and will use it as if it is a figure. \r\n" );
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document.write( "Let the point P be the intersection of diagonals AC and BD of the parallelogram.\r\n" );
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document.write( "The proof uses these properties from Geometry:\r\n" );
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document.write( "    1)  The straight line connecting midpoints of two sides of a triangle is parallel to the third side of the triangle.\r\n" );
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document.write( "    2)  If a straight line bisects one side of a triangle and is parallel to a second side of the triangle, then it bisects the third side of the triangle.\r\n" );
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document.write( "    3)  In a parallelogram, diagonals bisect each other.\r\n" );
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document.write( "    Regarding properties 1) and 2), see the lesson The line segment joining the midpoints of two sides of a triangle in this site.\r\n" );
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document.write( "    Regarding property 3), see the lesson Properties of diagonals of parallelograms in this site.\r\n" );
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document.write( "Based on Property 3), AP is half of AC.\r\n" );
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document.write( "Then AO is half of AP.\r\n" );
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document.write( "Then EO is the middle line of the triangle APD.\r\n" );
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document.write( "Hence, the line EO is parallel to DP   (Property 1).\r\n" );
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document.write( "Therefore, EF is parallel to DB.\r\n" );
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document.write( "Since EF is parallel to DB and E is the midpoint of AD, then the point F is the midpoint of AB, in accordance with the property 2).\r\n" );
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document.write( "The proof is completed.\r\n" );
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\n" ); document.write( "\n" ); document.write( "The problem is solved.\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook on Geometry\r
\n" ); document.write( "\n" ); document.write( " GEOMETRY - YOUR ONLINE TEXTBOOK \r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the parts of this online textbook under the topics \"Properties of triangles\" and \"Properties of parallelograms\".\r
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