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?
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
(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.
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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
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
Related material
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|Activities| |Contact| |Front page| |Contents| |Geometry| |Fallacies|
Copyright © 1996-2018 Alexander Bogomolny72002815