Rouse Ball's Fallacy
What Is Wrong?

According to J. dePillis, George Pólya used to define Geometry as the science of correct reasoning on incorrect figures. The quote in Eves is a little different: Geometry is the art of correct reasoning on incorrect figures, although the reference is exactly the same: How to Solve It?, 1945. (I have no way of verifying whose version is the correct one as, unfortunately, I could not locate the referenced quote in my 1973 edition. Might be missing the obvious.)

Either way, something is wrong with the diagram presented in the applet below. The reason I am so sure about that is because absolutely flawless reasoning based on that figure leads to an absurd result. The question is what is wrong?


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

The construction is as follows. Form a right angle ADC and an obtuse angle DAE (away from DC) so that DC = AE. Since the two segments EC and AD are not parallel, their perpendicular bisectors are not parallel either. Let them meet in point O. Let K and H be the midpoints of EC and DA, respectively. Then

(1)CO = EO, because ΔCEO is isosceles.
(2)DO = AO, because ΔADO is isosceles.
(3)DC = AE, by construction.

By the SSS criterion, ΔOCD = ΔOEA. Therefore,

(4)∠CDO = ∠EAO,

but also,

(5)∠ADO = ∠DAO.

Subtracting (5) from (4) yields

(5)90o = ∠ADC = ∠DAE.

We arrive at an absurd conclusion that the obtuse angle DAE is in fact right in contradiction with the construction.

What is wrong?

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This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?
 

Had the construction been 100% correct, the line EO would have lain outside the quadrilateral ADCE. In which case the subtraction (4) - (5) would have been meaningless. To get the fallacy, I shortened AE by 4%. This was enough to make EO pass inside the quadrilateral when AE was close to vertical.

References

  1. J. dePillis, 777 Mathematical Conversation Starters, MAA, 2002, p. 114
  2. H. W. Eves, Return to Mathematical Circles, PWS-KENT Publ Co, 1988, p. 79
  3. W.W. Rouse Ball and H.S.M. Coxeter, Mathematical Recreations and Essays, Dover, 1987

Related material
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Geometric Fallacies

  • A Circle With Two Centers
  • All Triangles Are Isosceles
  • Two Perpendiculars From a Point to a Line
  • Is Every Trapezoid Parallelogram?
  • Every Parallelogram Is a Rectangle
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