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>

Theorem
In an arbitrary convex quadrilateral the midpoints of its sides are vertices of the parallelogram. Prove.

<TABLE>
 <TR>
 <TD>
Proof

Let ABCD be an arbitrary convex quadrilateral (Figure 1),&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
and let the points E, F, G and H be the midpoints of its 
sides AB, BC, CD and AD respectively. 
We need to prove that the quadrilateral EFGH is the parallelogram.

Draw the diagonals AC and BD in the quadrilateral ABCD (Figure 2). 
The segment HG is the midpoint segment in the triangle ACD. 
Therefore, the segment HG is parallel to the side AC of the triangle 
ACD 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 Triangles of the section 
Geometry 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)) 
 )}}}
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 1. To the Theorem&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
 </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))
)}}}
Figure 2. To the proof of the Theorem
 </TD>
 </TR>
</TABLE>The segment EF is the midpoint segment in the triangle ABC. Therefore, the segment EF is parallel to the side AC of the triangle ABC. 
Since the segments HG and EF are both parallel to the diagonal AC, they are parallel to each other.

Similarly, the segment GF is the midpoint segment in the triangle DCB. Therefore, the segment GF is parallel to the side DB of the triangle DCB. 
The segment HE is the midpoint segment in the triangle ABD. Therefore, the segment HE is parallel to the side DB of the triangle ABD. 
Since the segments GF and HE are both parallel to the diagonal DB, they are parallel to each other.

Thus, we have proved that in the quadrilateral EFGH the opposite sides HG and EF, HE and GF are parallel by pairs.
Hence, the quadrilateral EFGH is the parallelogram. The Theorem is proved.

My other lessons on parallelograms in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Properties-of-the-sides-of-parallelograms.lesson>Properties of the sides of parallelograms</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Properties-of-diagonals-of-parallelograms.lesson>Properties of diagonals of parallelograms</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Opposite-angles-of-a-parallelogram-are-congruent.lesson>Opposite angles of a parallelogram</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Consecutive-angles-of-a-parallelogram.lesson>Consecutive angles of a parallelogram</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Parallelograms/Remarcable-advanced-problems-on-parallelograms.lesson>Remarcable advanced problems on parallelograms</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <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>

&nbsp;&nbsp;&nbsp;&nbsp;- <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 &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.