Lesson Midpoints of a quadrilateral are vertices of the parallelogram
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<H2>Midpoints of a quadrilateral are vertices of the parallelogram</H2> <B>Theorem</B> In an arbitrary convex quadrilateral the midpoints of its sides are vertices of the parallelogram. Prove. <TABLE> <TR> <TD> <B>Proof</B> Let <B>ABCD</B> be an arbitrary convex quadrilateral (<B>Figure 1</B>), and let the points <B>E</B>, <B>F</B>, <B>G</B> and <B>H</B> be the midpoints of its sides <B>AB</B>, <B>BC</B>, <B>CD</B> and <B>AD</B> respectively. We need to prove that the quadrilateral <B>EFGH</B> is the parallelogram. Draw the diagonals <B>AC</B> and <B>BD</B> in the quadrilateral <B>ABCD</B> (<B>Figure 2</B>). The segment <B>HG</B> is the midpoint segment in the triangle <B>ACD</B>. Therefore, the segment <B>HG</B> is parallel to the side <B>AC</B> of the triangle <B>ACD</B> in accordance to the lesson <A HREF=http://www.algebra.com/algebra/homework/Triangles/The-line-segment-joining-the-midpoints-of-two-sides-of-a-triangle.lesson>The line segment joining the midpoints of two sides of a triangle</A> (under the topic <B>Triangles</B> of the section <B>Geometry</B> in this site). </TD> <TD> {{{drawing( 240, 200, 0.5, 6.5, 0.5, 5.5, line( 1.0, 1.0, 6.0, 1.0), line( 2.0, 4.0, 5.0, 5.0), line( 1.0, 1.0, 2.0, 4.0), line( 6.0, 1.0, 5.0, 5.0), locate(1.0, 1.0, A), locate(6.0, 1.0, B), locate(5.0, 5.4, C), locate(2.0, 4.4, D), circle(3.5, 1.0, 0.05, 0.05), circle(3.5, 4.5, 0.05, 0.05), circle(1.5, 2.5, 0.05, 0.05), circle(5.5, 3.0, 0.05, 0.05), locate(3.5, 1.0, E), locate(5.6, 3.2, F), locate(3.4, 4.9, G), locate(1.2, 2.8, H), green(line (3.5, 1.0, 5.5, 3.0)), green(line (5.5, 3.0, 3.5, 4.5)), green(line (3.5, 4.5, 1.5, 2.5)), green(line (1.5, 2.5, 3.5, 1.0)) )}}} <B>Figure 1</B>. To the <B>Theorem</B> </TD> <TD> {{{drawing( 240, 200, 0.5, 6.5, 0.5, 5.5, line( 1.0, 1.0, 6.0, 1.0), line( 2.0, 4.0, 5.0, 5.0), line( 1.0, 1.0, 2.0, 4.0), line( 6.0, 1.0, 5.0, 5.0), locate(1.0, 1.0, A), locate(6.0, 1.0, B), locate(5.0, 5.4, C), locate(2.0, 4.4, D), circle(3.5, 1.0, 0.05, 0.05), circle(3.5, 4.5, 0.05, 0.05), circle(1.5, 2.5, 0.05, 0.05), circle(5.5, 3.0, 0.05, 0.05), locate(3.5, 1.0, E), locate(5.6, 3.2, F), locate(3.4, 4.9, G), locate(1.2, 2.8, H), green(line (3.5, 1.0, 5.5, 3.0)), green(line (5.5, 3.0, 3.5, 4.5)), green(line (3.5, 4.5, 1.5, 2.5)), green(line (1.5, 2.5, 3.5, 1.0)), red(line(1.0, 1.0, 5.0, 5.0)), red(line(2.0, 4.0, 6.0, 1.0)) )}}} <B>Figure 2</B>. To the proof of the <B>Theorem</B> </TD> </TR> </TABLE>The segment <B>EF</B> is the midpoint segment in the triangle <B>ABC</B>. Therefore, the segment <B>EF</B> is parallel to the side <B>AC</B> of the triangle <B>ABC</B>. Since the segments <B>HG</B> and <B>EF</B> are both parallel to the diagonal <B>AC</B>, they are parallel to each other. Similarly, the segment <B>GF</B> is the midpoint segment in the triangle <B>DCB</B>. Therefore, the segment <B>GF</B> is parallel to the side <B>DB</B> of the triangle <B>DCB</B>. The segment <B>HE</B> is the midpoint segment in the triangle <B>ABD</B>. Therefore, the segment <B>HE</B> is parallel to the side <B>DB</B> of the triangle <B>ABD</B>. Since the segments <B>GF</B> and <B>HE</B> are both parallel to the diagonal <B>DB</B>, they are parallel to each other. Thus, we have proved that in the quadrilateral <B>EFGH</B> the opposite sides <B>HG</B> and <B>EF</B>, <B>HE</B> and <B>GF</B> are parallel by pairs. Hence, the quadrilateral <B>EFGH</B> is the parallelogram. The <B>Theorem</B> is proved. My other lessons on parallelograms in this site are - <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/In-a-parallelogram-each-diagonal-divides-it-in-two-congruent-triangles.lesson>In a parallelogram, each diagonal divides it in two congruent triangles</A> - <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Properties-of-the-sides-of-a-parallelogram.lesson>Properties of the sides of a parallelogram</A> - <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Properties-of-the-sides-of-parallelograms.lesson>Properties of the sides of parallelograms</A> - <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Properties-of-diagonals-of-parallelograms.lesson>Properties of diagonals of parallelograms</A> - <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Opposite-angles-of-a-parallelogram-are-congruent.lesson>Opposite angles of a parallelogram</A> - <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Consecutive-angles-of-a-parallelogram.lesson>Consecutive angles of a parallelogram</A> - <A HREF=http://www.algebra.com/algebra/homework/word/geometry/The-length-of-diagonals-of-a-parallelogram.lesson>The length of diagonals of a parallelogram</A> - <A HREF=https://www.algebra.com/algebra/homework/Parallelograms/Remarcable-advanced-problems-on-parallelograms.lesson>Remarcable advanced problems on parallelograms</A> - <A HREF=http://www.algebra.com/algebra/homework/word/geometry/How-to-solve-problems-on-the-parallelogram-sides-measures-Examples.lesson>HOW TO solve problems on the parallelogram sides measures - Examples</A> - <A HREF=http://www.algebra.com/algebra/homework/word/geometry/How-to-solve-problems-on-the-angles-of-parallelograms.lesson>HOW TO solve problems on the angles of parallelograms - Examples</A> - <A HREF=https://www.algebra.com/algebra/homework/Parallelograms/PROPERTIES-OF-PARALLELOGRAMS.lesson>PROPERTIES OF PARALLELOGRAMS</A> To navigate over all topics/lessons of the Online Geometry Textbook use this file/link <A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.