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I'm teaching a geometry course this semester, involving mainly Euclidean geometry and introducing non-Euclidean geometry. In discussing the importance of deductive proof, I'd like to present some examples of statements that may appear to be true (perhaps based on a common student misconception or over-generalisation), but are not. The aim would be to reinforce one of the reasons given for studying deductive proof: to properly determine the validity of statements claimed to be true.

Can anyone offer interesting examples of such statements?

An example would be that the circumcentre of a triangle lies inside the triangle. This is true for triangles without any obtuse angles - which seems to be the standard student (mis)conception of a triangle. However, I don't think that this is a particularly good example because the misconception is fairly easily revealed, as would statements that hold only for isoceles or right-angled triangles. I'd really like to have some statements whose lack of general validity is quite hard to tease out, or has some subtlety behind it.

Of course the phrase 'may appear to be true' is subjective. The example quoted should be understood as indicating the level of thinking of the relevant students.

Thanks.

user12477
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    Not completely relevant, but related: http://math.stackexchange.com/questions/111440/examples-of-apparent-patterns-that-eventually-fail. – tomasz Sep 22 '12 at 19:28
  • Would you consider statements that are false but become true when adding genericity hypotheses? There are lots of geometric statements that fail or become meaningless if the initial data are singular in some way. – Marc van Leeuwen Sep 22 '12 at 19:34
  • This answer to that question seems directly relevant to me. – celtschk Sep 22 '12 at 19:35
  • Thanks for the pointer @tomasz - there are a few good examples there. – user12477 Sep 22 '12 at 19:46
  • @MarcvanLeeuwen - I guess it depends. For example, "the square on one side is not equal to the sum of the squares on the other two sides" falls into this category, but I'd think of this as falling into the 'readily resolved' category. (Which is perhaps a bit fussy of me!) – user12477 Sep 22 '12 at 19:50
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    I meant something like "In a triangle $ABC$, the image of the orthocenter by reflection in $BC$ is the point distict from $A$ where the altitude of $A$ intersects the circumcircle". The part "distinct from $A$" is important, and visually obvious, and fails in rare circumstances (when the altitude is tangent to the circumcircle). – Marc van Leeuwen Sep 23 '12 at 04:41
  • @Marc - Nice example: while it's not something that would necessarily occur to students, it seems 'right' that the construction should lead to a point distinct from A. – user12477 Sep 27 '12 at 23:01

3 Answers3

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Also trivial but maybe interesting: two sides and an angle determine a triangle up to congruence.

Nate Eldredge
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  • Uhm... isn't this one true? – akkkk Sep 22 '12 at 19:52
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    @Auke - depending on which angle! – Belgi Sep 22 '12 at 19:54
  • Trivial is good. Kind of more about the importance of precise language than about concepts, but certainly would trip up students by invoking the SAS mantra! You got me - I was on the verge of making @Auke's comment... – user12477 Sep 22 '12 at 20:02
  • @user12477 I'm not sure she ever wrote it down or even said it out loud, but my teacher indicated this was the ASS axiom, which is a good way to remember, at least if English is your first language. – Will Jagy Sep 22 '12 at 21:14
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    @Will :) I've always understood the order SAS (ASS, ASA,...) to indicate that in XYZ, the object Y is positioned between the objects X and Z. So if your teacher described the axiom as you've said, she was talking out her angle-side-side... – user12477 Sep 22 '12 at 21:49
  • @user12477, there you go then. I've always been able to remember that this one is false. – Will Jagy Sep 22 '12 at 21:54
  • @Nate - very late with accepting this, but returning now because this suggestion worked well with students. – user12477 Nov 01 '12 at 22:31
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Here is one example that is quite similar in nature to the statement in the question about the center of the circumcircle lying inside a triangle, but the dubious part ("lie inside") is somewhat better disguised. I report it only because I just found it in Wikipedia, with a literature reference.

The incenter (that is, the center for the inscribed circle) of the orthic triangle is the orthocenter of the original triangle.

It is easily verified that for a triangle with an obtuse angle the orthocenter lies outside the orthic triangle, so it cannot be the incenter in this case; it is one of the excenters instead.

Marc van Leeuwen
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A common attempt to trisect an angle is to construct an isosceles triangle, $\triangle ABC$, with $\angle ABC$ the desired angle; then trisect the opposite side $\overline{AC}$, finding points $D,E$ on $\overline{AC}$ with $AD = DE = EC$. One might guess that $m\angle ABD = m\angle DBE = m\angle EBC$ but in fact this is never the case.

Although this isn't quite what you were asking for, the best example of a subtly flawed geometric proof I know of can be seen at Wikipedia here.

Jair Taylor
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