
Parallelogram inscribed in a quadrilateral
Try this
Drag any orange dot and note that the red lines always form a parallelogram.
If you find the
midpoints
of each side of any
quadrilateral,
then link them sequentially with lines, the result is always a
parallelogram.
This may seem unintuitive at first, but if you drag any vertex of the quadrilateral above,
you will see it is in fact always true, even when the quadrilateral is 'selfcrossing'  where some sides of the quadrilateral cross over other sides.
Proof
The figure below is the same as above, except with the points J,K,L, M labelled and the line DB added.
By definition J,K,L,M are the midpoints of their respective sides.

Argument 
Reason 
1 
JM is the midsegment of the triangle ABD 
The midsegment of a triangle is a line linking the midpoints of two sides (See
Midsegment of a triangle) 
2 
JM is half DB and parallel to it 
From the properties of the midsegment of a triangle 
3 
Likewise in triangle DBC, LK is also half DB and parallel to it 
From the properties of the midsegment of a triangle 
4 
JKLM is a parallelogram 
A pair of opposite sides (LK and JM) are
parallel
and
congruent 
 Q.E.D
Other polygon topics
General
Types of polygon
Area of various polygon types
Perimeter of various polygon types
Angles associated with polygons
Named polygons
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