I used <a href="http://www.usna.edu/Users/cs/qepcad/B/QEPCAD.html">qepcad</a> to verify that there are no convex pentagons for which this ratio of areas is 1/2 or 3/4. Since the set of possible areas is the image of a connected space under a continuous function, and we know the set contains (1/2, 3/4), it must equal (1/2, 3/4). Here is a log of the qepcad session.
<pre>
=======================================================
Quantifier Elimination
in
Elementary Algebra and Geometry
by
Partial Cylindrical Algebraic Decomposition
Version B 1.53, 16 Jul 2009
by
Hoon Hong
([email protected])
With contributions by: Christopher W. Brown, George E.
Collins, Mark J. Encarnacion, Jeremy R. Johnson
Werner Krandick, Richard Liska, Scott McCallum,
Nicolas Robidoux, and Stanly Steinberg
=======================================================
Enter an informal description between '[' and ']':
<b>[ area of middle pentagon ]</b>
Enter a variable list:
<b>(a,x1,y1,x2,y2)</b>
Enter the number of free variables:
<b>1</b>
Enter a prenex formula:
<b>(E x1)(E y1)(E x2)(E y2)[ [ a = 1/2 \/ a = 3/4 ] /\
x1 > 0 /\ y1 > 0 /\ 1 - x1 - y1 < 0 /\
x2 > 0 /\ x2 y1 + y2 - x1 y2 - y1 < 0 /\ x1 + x2 y1 - x2 - x1 y2 < 0 /\
a (1/2)(y1 + x1 y2 - x2 y1 + x2) = (1/8)(0 - 1 + x1 + 2 x2 + 2 y1 + y2 + 2 x1 y2 - 2 x2 y1) ].</b>
=======================================================
Before Normalization >
<b>finish</b>
An equivalent quantifier-free formula:
FALSE
===================== The End =======================
-----------------------------------------------------------------------------
3 Garbage collections, 1229018 Cells and 0 Arrays reclaimed, in 24 milliseconds.
98190 Cells in AVAIL, 500000 Cells in SPACE.
System time: 292 milliseconds.
System time after the initialization: 264 milliseconds.
-----------------------------------------------------------------------------
</pre>