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Thread: Hierarchical Quadrilateral Tree

  1. #1

    Red face Hierarchical Quadrilateral Tree

    I've posted a Hierarchical Quadrilateral Tree sketch under my Files (a modified & slightly extended version of the one posted earlier by user cooperti09). It shows the hierarchical relationships between several quadrilaterals as well as (several of) their properties. General cases can be dragged into special cases.

    In a dynamic geometry environment, 'partition' ('apartheid') thinking that, for example, excludes the parallelograms from the trapezoids is simply not sensible when one can dynamically drag a trapezoid to become a parallelogram. More-over, to exclude these special cases is just not convenient mathematically for a number of reasons, discussed more fully in the links & resources below.

    Links to other related online activities and readings are provided. For more background information on the usefulness of a hierarchical classification of quadrilaterals, as well as proofs of several of the given properties, please consult my books Rethinking Proof with Sketchpad at http://www.keypress.com/x5588.xml and/or Some Adventures in Euclidean Geometry at http://www.lulu.com/content/7622884

    Hope teachers will find the sketch useful & comments are welcome!

    Last edited by Michael de Villiers; 12-10-2010 at 01:10 PM.

  2. #2
    Michael's post is at http://sketchexchange.keypress.com/s...rilateral-tree and he has kindly provided both Sketchpad 4 and 5 versions. Thank you!

  3. #3
    Thanks, Elizabeth for posting a direct link!

    Perhaps to quickly mention that one of the advantages of a hierarchical classification of quadrilaterals, as opposed to a partition ('apartheid') one, is that theorems for general quadrilaterals then do not need to be reproved for special cases. For example, proving that the diagonals of a parallelogram bisect each other, means that it immediately applies to all the special cases. In a partition classification, however, such as Euclid actually used in his Elements (300 BC) by excluding the squares from both the rectangles and rhombi, and the rectangles and the rhombi from the parallelograms, the number of duplicated proofs increases drastically.

    The same applies for the trapezoid and parallelogram, and in the GSP sketch some properties (theorems) of trapezoids have already been listed that also apply to parallelograms. Here's another two interesting properties/theorems of trapezoids that also apply to parallelograms and isosceles trapezoids:
    1) the diagonals intersect on the line connecting the midpoints of the parallel sides
    2) If AB and CD are the parallel sides of ABCD, then BD^2 + AC^2 = AD^2 + BC^2 + 2AB*CD.
    Last edited by Michael de Villiers; 12-10-2010 at 01:07 PM.

  4. #4
    Removed as information is dated.
    Last edited by Michael de Villiers; 03-26-2011 at 12:17 PM.

  5. #5
    Just to let users know that I've added a 'Bisecting Quadrilateral' to the Quadrilateral Tree sketch posted earlier at http://sketchexchange.keypress.com/s...rilateral-tree and made a few other minor edits.

  6. #6
    I've now created an online, interactive version of this Hierarchical Quadrilateral Tree, and which has four more special quadrilaterals. It can be accessed on my Dynamic Geometry Sketches site at: http://math.kennesaw.edu/~mdevilli/quad-tree-web.html

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