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The importance of characterisations in geometry

Published online by Cambridge University Press:  18 June 2018

Martin Josefsson
Martin Josefsson*
Affiliation:
Västergatan 25d, 285 37 Markaryd, Sweden e-mail: [email protected]
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Extract

When studying geometry, it is natural to choose an object and try to determine what properties it has. Take for example a trapezium (trapezoid in American English): a quadrilateral with at least one pair of opposite parallel sides. Then we probably want to know how to calculate its area, the length of the diagonals, what other properties the diagonals might have, if it has some angle properties, how to calculate the length of the midsegment and so on. All these are interesting investigations. But when they are done (and in many textbooks the story is not allowed to continue even this far), then it is very common to consider the exploration of the trapezium finished. The truth is that here we are only half way. The most interesting part of the journey is the second half. So far we only know necessary conditions in the trapezium, that is, given that we have a quadrilateral that is a trapezium, what properties it has. But what about the reverse question? Which of these properties are unique to the trapezium, that is, which are the sufficient conditions? Or in other words: which properties characterise trapezia and separate them from all the other quadrilaterals? To me that is the really interesting question!

Type
Matter for Debate
Information
The Mathematical Gazette , Volume 102 , Issue 554 , July 2018 , pp. 302 - 307
Copyright
Copyright © Mathematical Association 2018 

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References

1. Josefsson, M., The area of the diagonal point triangle, Forum Geom. 11 (2011) pp. 213–216.Google Scholar
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3. Gardiner, A. D. and Bradley, C. J., Plane Euclidean geometry: theory and problems, The United Kingdom Mathematics Trust (2005).Google Scholar
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6. Josefsson, M., Similar metric characterizations of tangential and extangential quadrilaterals, Forum Geom. 12 (2012) pp. 63–77.Google Scholar
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