What's the appropriate education level for the following concise but non-trivial geometry problem?

Points $A$, $B$, $C$ are collinear; $\|AB\|=\|BD\|=\|CD\|=1$; $\|AC\|=\|AD\|$.
What is the set of possible $\|AC\|$ ?

To check one's answer, hover mouse over the following very partial spoiler.

The mean of the elements of the set of solutions is $\approx 1.08$.

Link to near-complete spoiler illustration; ctrl-click for new window or tab.

  • $\begingroup$ @BenjaminDickman Thanks for the suggestion. I have done so. $\endgroup$
    – Jim Belk
    Apr 17, 2014 at 2:39
  • $\begingroup$ I get a mean of ~1.118 instead. $\endgroup$
    – user173
    Apr 17, 2014 at 4:02
  • $\begingroup$ @MattF. I stand by my mean. Despite its simplicity, this problems requires rigor. The illustration was made for a simplified statement aiming at a young audience, using Alice, Bob, Carol and David rather than points, and is only NEAR-complete for the present statement. $\endgroup$
    – fgrieu
    Apr 17, 2014 at 7:56
  • $\begingroup$ OK, now I see your trick. $\endgroup$
    – user173
    Apr 17, 2014 at 11:51

3 Answers 3


I disagree with the other two answers. I would assign this problem in an introductory high school class, but I would not use it as an assessment item.

Here's why:

  • Most high school students will understand what this problem is asking.
  • There is some evidence that suggests that struggling with a problem, when one is very clear about the expectations of the problem, improves ones ability to solve problems and importantly, to remember the information required to solve the problem later.
  • Too often lower to mid-level students are given problems which do not require them to think. If we don't ask these students to think, we cannot expect them to improve their ability to do so.
  • This problem is what I would call an open middle problem. It is relatively easily stated, and has a single possible answer, but in the middle students can approach it in a variety of different ways. This will help prompt students, particularly if they work in groups, to share different strategies for approaching the problem with each other, and to get feedback on those strategies. Even if students don't get the right answer, they will learn from this experience.
  • 1
    $\begingroup$ I wouldn't use this problem because the cute set-up makes it artificially hard to draw the possible configurations. For open-middle problems, I would prefer What is the largest area of a quadrilateral that fits between concentric circles of radius 1 and 2? or If an equilateral triangle has side 1, what are the possible side lengths of a square that touches it at exactly two points? $\endgroup$
    – user173
    Apr 17, 2014 at 21:30
  • $\begingroup$ @Matt F.: I do not get what you mean with "the cute set-up makes it artificially hard to draw the possible configurations". I intend to make the statement a concise 2-liner, with unambiguous indication that all solutions are wanted but no hint of how many there are, in order to promote and better reward rigor in the reasoning. I used set to avoid is/are (I'm not a native English speaker and perhaps there is a better formulation towards that goal?). There was no intention to mislead in the question around the problem statement (the spoiler illustration was pre-existing and hard to fix). $\endgroup$
    – fgrieu
    Apr 25, 2014 at 21:30
  • $\begingroup$ @fgrieu OK, I replace "artificially hard" with "unnecessarily hard for my taste". $\endgroup$
    – user173
    Apr 26, 2014 at 2:45

It depends on how difficult you want the problem to be. This seems like it would be a good "math contest" type problem for high-school students. For example, this could easily be on the AMC 10 or AMC 12. It might also work well as a longer-term or "project" type homework assignment in a geometry or trigonometry class.


I agree with Jim Belk that this would be a good "math contest" type problem for high-school students. For example, this could easily be on the AMC 10 or AMC 12.

I would not assign the problem in an ordinary class, where the difficulty in checking the answer would seem to promote more frustration than payoff.


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