Question 832299
Consider quadrilateral ABCD. If the midpoint of AB is E, the midpoint of BC is F, the midpoint of CD is G, the midpoint of AD is H. What conclusions can you make about quadrilateral EFGH? Justify.

EXPLORE for at least three different quadrilaterals of the following six: Quadrilateral with no special properties, parallelogram, rectangle, rhombus, square, trapezoid.  Justifications can include measuring sides and angles

Frankie
<pre>

{{{drawing(300,210,-1,9,-1,6,
red(line(1,2,4,0),line(4,0,7.5,2.5),line(7.5,2.5,4.5,4.5),line(4.5,4.5,1,2)),
locate(.6,2.35,E),locate(4,0,F),locate(7.7,2.8,G),locate(4.4,5.1,H),
locate(-.2,0,B),locate(8,0,C),locate(7.1,5.4,D),locate(1.8,4.5,A),
line(0,0,8,0),line(8,0,7,5),line(7,5,2,4),line(0,0,2,4)  )}}}

Draw diagonals AC and BD 

{{{drawing(300,210,-1,9,-1,6,
red(line(1,2,4,0),line(4,0,7.5,2.5),line(7.5,2.5,4.5,4.5),line(4.5,4.5,1,2)),
locate(.6,2.35,E),locate(4,0,F),locate(7.7,2.8,G),locate(4.4,5.1,H),
locate(-.2,0,B),locate(8,0,C),locate(7.1,5.4,D),locate(1.8,4.5,A),
line(0,0,8,0),line(8,0,7,5),line(7,5,2,4),line(0,0,2,4),
green(line(2,4,8,0))
  )}}}{{{drawing(300,210,-1,9,-1,6,
red(line(1,2,4,0),line(4,0,7.5,2.5),line(7.5,2.5,4.5,4.5),line(4.5,4.5,1,2)),
locate(.6,2.35,E),locate(4,0,F),locate(7.7,2.8,G),locate(4.4,5.1,H),
locate(-.2,0,B),locate(8,0,C),locate(7.1,5.4,D),locate(1.8,4.5,A),
line(0,0,8,0),line(8,0,7,5),line(7,5,2,4),line(0,0,2,4),
green(line(0,0,7,5))
  )}}}
In the figure on the left:
HG is the midsegment of &#916;DAC so HG&#8741;AC
EF is the midsegment of &#916;ABC so EF&#8741;AC
So EF&#8741;AC, since lines parallel to the same line 
are parallel to each other.

In the figure on the right:
EH is the midsegment of &#916;ABD so EH&#8741;BD
FG is the midsegment of &#916;DBC so FG&#8741;BD

So EH&#8741;FG, since lines parallel to the same line 
are parallel to each other. 

So EFGH is a parallelogram because both pairs of opposite sides are
parallel.

I'll finish this later.
Edwin</pre>