Assignment #6
Nicole Mosteller
EMAT 6680

The key the relationships of these triangles is that the point H is the midpoint of segment EJ and segment FC.
See below for the proof of point H as midpoint as well as the relationships that follow.

Remember that parallel lines have been constructed to arrive at point J.  
Since segment BC is parallel to segment FJ, we know that 
<FJE = <JEC (alt. interior angles).
Also using these parallel lines and that segment BF is parallel to segment EJ,
we know that quad. BFJE is a parallelogram with opposite sides congruent.
This gives segments FJ = BE.
Since segment AE is a median of DABC, we know that segments BE = CE.
From the transitive property, we know that
Segments FJ = CE.
Using the vertical angles at the point H, we see that DECH = DJFH (AAS).

Now see the relationships that come from these congruent triangles.
Segments EH = JH (CPCTC).
Segment AH is a median of DAEJ.
Now from the lemma on the previous page, 
Area of DAHJ = Area of DEAH.

Segments CH = FH (CPCTC).
Segment EH is a median of DCEF, and
 Area of DCEH = Area of DHEF.

The original construction of medians gave segment BF a median of DABC.
Since point F is a midpoint of segment AC, we know that
Segment EF is a median of DCAE, and
Area of DCEF = Area of DAEF.

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