# Rouse Ball's FallacyWhat 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 https://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,

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? ### 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 https://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 • 