# 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) | 90^{o} = ∠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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Copyright © 1996-2018 Alexander Bogomolny

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

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Copyright © 1996-2018 Alexander Bogomolny