An Unexpected Fraction

   
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2016/09/18 - An Unexpected Fraction

On 2016-09-09, @MEIMaths tweeted an image and said:

http://www.solipsys.co.uk/images/QuadrilateralProblem_Question.png
Quadrilateral Problem

An Unexpected Fraction?

Start with any convex quadrilateral. Mark the midpoint of each side, join these midpoints to the vertex two places clockwise around the quadrilateral.

What fraction of the original quadrilateral is the new quadrilateral?

The way the question is phrased makes you feel two things.

Firstly, you have to feel that the answer is likely to be surprising in some way, so that means it's likely to be a nice puzzle.

But secondly, it seems clear from the phrasing that the answer must be the same regardless of the particular quadrilateral chosen.

Well, I've seen this before in the context of a square, and the answer there is lovely - it's a fifth. The proof in the case of a square is fairly simple, and there are a number of ways of going about it. So there we are, it's sorted, the answer is a fifth.

Nice one.

http://www.solipsys.co.uk/images/QuadrilateralProblem_Square.png
Quadrilateral Problem
The Square case
But then Hugh Hunt sent me an email saying:

"In the limiting case of
an equilateral triangle
the answer is a sixth.

"Am I doing something
wrong?"

http://www.solipsys.co.uk/images/QuadrilateralProblem_Triangle.png
Quadrilateral Problem
Limiting Triangle
That's a good question. If we take two adjacent vertices of the square and move them together, just before they meet we do still have a convex quadrilateral, so because we expect the answer to be constant (because of the context) we would expect the answer still to be a fifth, but the answer is now pretty obviously a sixth.

So what's going on?

The short answer seems to be that @MEIMaths mis-spoke themselves, and despite the wording, they didn't actually mean any quadrilateral. But what might they have meant, and what answers might be possible?

So here are some interesting questions:

  • Under what circumstances is the answer one fifth?

  • Is the answer always between a sixth and a fifth?

    • If not, what are the extreme values?

  • Is every value in the range $[\frac{1}{6},\frac{1}{5}]$ achievable?

So even if it was a mistake, there is interesting mathematics to be found. That, at least, is no surprise.


References

Here's the original tweet:

Adam Atkinson has created a GeoGebra twitchet you can play with:

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