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>