What Is Wrong?
(Rouse Ball's Fallacy)
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?
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 CEO is isosceles. |
| (2) | DO = AO, because ADO is isosceles. |
| (3) | DC = AE, by construction. |
By the SSS criterion, OCD = OEA. 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?
Copyright © 1996-2008 Alexander Bogomolny
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
- J. dePillis, 777 Mathematical Conversation Starters, MAA, 2002, p. 114
- H. W. Eves, Return to Mathematical Circles, PWS-KENT Publ Co, 1988, p. 79
- W.W. Rouse Ball and H.S.M. Coxeter, Mathematical Recreations and Essays, Dover, 1987
Copyright © 1996-2008 Alexander Bogomolny
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