Some Trapezoid (Trapezium) Explorations

Below are given 5 different dynamic explorations for a trapezoid. Scroll down to access them and click on the corresponding buttons in the sketch below. In each case, check whether the result is also true for special cases like a parallelogram or a crossed quadrilateral - also see the Concluding Remark at the bottom of this page.

1. Midpoint Trapezium Theorem
Given any (convex) trapezium ABCD with AD // BC, and E and F the respective midpoints of opposite sides AB and CD. Then EF = (AD + BC)/2.
1) What do you notice about EF and (AD + BC)/2?
2) Drag the trapezium to check your observation above. Also check for special cases like a parallelogram (as well as a crossed case of the trapezium).
Challenge
3) Can you explain why (prove) this theorem is true? Can you prove it in more than one way?
Explore More
4) What happens to the above relationship if ABCD is any quadrilateral?
5) Explore with your own dynamic sketch or go here.

Web Sketchpad

Some Trapezium Explorations

2. Trapezium Equal Segments Investigation
6) Click on the 'Link to equal segments' button on the bottom right to navigate to a new sketch.
This dynamic sketch represents any trapezium (trapezoid) ABCD with AD // BC and straight line FEG drawn parallel to AD and BC.
7) What do you notice about EF and EG?
8) Write down your conjecture in the form "If ..., then ..."
9) Drag the trapezium to check your observation above. Also check for special cases like a parallelogram (as well as a crossed case of the trapezium).
Challenge
10) Can you explain why (prove) your observation is true? Can you prove it in more than one way?

3. Midpoint Collinearity Investigation
11) Click on the 'Link to midpoint collinearity' button on the bottom right to navigate to a new sketch.
This dynamic sketch represents any trapezium (trapezoid) ABCD with AD // BC, with F and G the respective midpoints of BC and AD, and E the intersection of the diagonals.
12) What do you notice about the points E, F and G?
13) Write down your conjecture in the form "If ..., then ..."
14) Drag the trapezium to check your observation above. Also check for special cases like a parallelogram (as well as a crossed case of the trapezium).
Challenge
15) Can you explain why (prove) your observation is true? Can you prove it in more than one way?
Hint: One easy way to prove it is to extend the non-parallel sides to meet in a point to form a triangle, and then consider the medians of the two formed similar triangles, as well as those formed by the diagonals. (The parallelogram case basically follows from symmetry).
Note: This result is known as Steiner's Trapezium Theorem and is named after the famous Swiss geometer, Jakob Steiner (1796–1863).

4. Squares Collinearity Investigation
16) Click on the 'Link to squares collinearity' button on the bottom right to navigate to a new sketch.
This dynamic sketch shows any trapezium (trapezoid) ABCD with AD // BC with squares with centres F and G, respectively constructed on sides BC and AD, and E the intersection of the diagonals.
17) What do you notice about the points E, F and G?
18) Write down your conjecture in the form "If ..., then ..."
19) Drag the trapezium to check your observation above. Also check for special cases like a parallelogram (as well as a crossed case of the trapezium).
Challenge
20) Can you explain why (prove) your observation is true? Can you prove it in more than one way?
Further Generalization
21) Can you generalize this result further to other similar quadrilaterals on the sides?

5. Perpendicular Bisectors form Similar Trapezoid Investigation
22) Click on the 'Link to similar trapezoid' button on the bottom right to navigate to a new sketch.
This dynamic sketch shows any trapezium (trapezoid) ABCD with AD // BC with the perpendicular bisectors of the sides constructed to form another quadrilateral EFGH.
23) What do you notice about the quadrilateral EFGH in relation to ABCD?
24) Click on the 'Show Measurements' button.
25) Write down your conjecture in the form "If ..., then ..."
26) Drag the trapezium to check your observation above. Also check for special cases like a parallelogram (as well as a crossed case of the trapezium).
Challenge
27) Can you explain why (prove) your observation is true? Can you prove it in more than one way?

Notes
a) The first two investigations often appear in high school texts around the world.
b) The third investigation appears less often in high school texts, probably because it is slightly more challenging.
c) A proof of the Squares Collinearity result is given on p. 168 of my Some Adventures in Euclidean Geometry and generalizations to similar rectangles and rhombi on p. 187. The book is available to download as a free PDF at Some Adventures in Euclidean Geometry.
d) Surprisingly, the last investigation does not appear to be well-known, though it does appear in Radko & Tsukerman (2012, p. 166); Humenberger & Schuppar (2024).

Concluding Remark
As you should've observed in all 5 the investigations above, each one of the results also hold when ABCD becomes a parallelogram. It is therefore much more convenient and economical to define a trapezoid (trapezium) as 'a quadrilateral with at least one pair of opposite sides parallel' so as to include the parallelograms. If trapezoids (trapeziums) are defined in such a way as to exclude parallelograms as is sometimes done, one would in each case have to do another separate proof for a parallelogram as well (compare with Josefsson, 2016, pp. 75-78; Usiskin et al, 2008; pp. 29-32; De Villiers, 2025). The problem is that once students have been exposed to an exclusive definition of a trapezoid (trapezium) this conceptualization tends to fossilize and is later likely to cause cognitive conflict that is often difficult to resolve and resistant to change (Budiarto, 2018). A study of preservice teachers by Erdogan & Dur (2014) also found that the majority of these preservice teachers had difficulty defining a trapezoid (trapezium) in an inclusive way.

Some References
Budiarto, M.T. (2018). Fauzi’s cognitive conflict in the development of geometry teaching material: A case study in shifting trapezoidal definition. Journal of Physics: Conference Series, 1088 012083.
De Villiers, M. (2025). ‘Apartheid’ Definitions of Trapezia Must Fall!. Learning and Teaching Mathematics, No. 38, pp. 40-43.
Erdogan, E.O. & Dur, Z. (2014). Preservice Mathematics Teachers’ Personal Figural Concepts and Classifications About Quadrilaterals. Australian Journal of Teacher Education, Vol. 39, Issue 6, pp. 107-133.
Humenberger, H. & Schuppar, B. (2024). Anschauliche Elementargeometrie (in German). SpringerSpektrum, Berlin.
Josefsson, M. (2016). On the classification of convex quadrilaterals. The Mathematical Gazette, Vol. 100, No. 547, pp. 68-85.
Radko, O. & Tsukerman, E. (2012). The Perpendicular Bisector Construction, the Isoptic Point and the Simson Line of a Quadrilateral. Forum Geometricorum, Vol 12, pp. 161-189.
Usiskin, Z., Griffin, J., Witonsky, D., & Willmore, E. (2008). The classification of quadrilaterals: a study of definition. Charlotte: Information Age Publishing.

External Links
SA Mathematics Olympiad
Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Created by Michael de Villiers in 2 Feb 2014; converted to WebSketchpad, 27 Feb 2025; updated 2/9 March 2025; 22 May 2025.