I used <a href="http://www.usna.edu/Users/cs/qepcad/B/QEPCAD.html">qepcad</a> to compute that the intersection of the set of possible area ratios with the interval [1/2, 3/4] is (1/2, 3/4). Since the set of possible area ratios is the image of a connected space under a continuous function, and we know the set contains (1/2, 3/4), but not 1/2 or 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:
2 a - 1 > 0 /\ 4 a - 3 < 0
===================== The End =======================
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12 Garbage collections, 473385670 Cells and 0 Arrays reclaimed, in 8158 milliseconds.
1345504 Cells in AVAIL, 40000000 Cells in SPACE.
System time: 79624 milliseconds.
System time after the initialization: 79028 milliseconds.
-----------------------------------------------------------------------------
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