The dynamic geometry activities below are from my book Rethinking Proof (free to download).
Worksheet & Teacher Notes
Open (and/or download) a guided worksheet and teacher notes to use together with the dynamic sketch below at: Worksheet & Teacher Notes: Reasoning Backwards: Triangle Midpoints.
"No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if it has affected teaching matter, is the didactical inversion, which as it happens may be anti-didactical." - Hans Freudenthal (1961) in The Concept and the Role of the Model in Mathematics and Natural and Social Sciences, Utrecht, D. Reidel Publishing Company.
Reasoning Backwards: Triangle Midpoints
In the earlier Kite Midpoints activity as well as the Isosceles Trapezoid Midpoints activity (not yet converted to WebSketchpad), we used the result that a line segment connecting the midpoints of two sides of a triangle is parallel to the third side (as well as equal to half its length).
There we used it as an assumption without any proof.
But why is it true that a line segment connecting the midpoints of two sides of a triangle is parallel to the third side (as well as equal to half its length)? Can we also prove it?
The activity below, and associated worksheet (see link above), deals with these questions.
Prerequisites: Kite Midpoints, Isosceles Trapezoid Midpoints (not yet converted to WebSketchpad), and Logical Discovery (Varignon Parallelogram Perimeter) activities in this book, and knowledge of the properties of and conditions for a parallelogram.
Reasoning Backwards: Triangle Midpoints
Notes
1) This worksheet focuses on the systematization function of proof, since we are here constructing a proof for a result that was earlier discovered and accepted without proof. With the traditional deductive approach, this result and its proof would be presented before its application to results such as Varignon’s theorem and kite and isosceles trapezoid midpoints. However, in actual mathematical research, results are seldom discovered in this straightforward linear fashion. For example, we might first discover an interesting result (for example, Varignon’s theorem) and then, upon trying to prove it, find that it can be proved in terms of another result (triangle midpoints). Our attention then shifts to proving this other result (the lemma, if you like).
'...........In writing up the results and their proofs, it is of course conventional to first prove the lemma and then the main result, but if this is (always) used as a teaching approach, it hides the fact that the actual sequence of discovery may have been the other way around. This worksheet attempts to give students some insight into the way a deductive ordering of some results may be arrived at by reasoning backward, rather than pretending that we always have the phenomenal foresight to first prove a particular, relatively uninteresting theorem (or lemma) because we anticipate that it will be used in proving important, interesting results later on. (For more details, also see De Villiers (1986, 1987)).
2) It is good pedagogy to perhaps first let students strugle with the problem statement a bit without giving them the construction hint. In my experience, most students (unless they know or remember an earlier proof) struggle to solve it as there is little to work or reason with in the stated problem. Good advice to give students then is the always useful problem solving heuristic geometry to try and draw auxilliary lines, and then to look for ways to relate it to known results. Let them perhaps first try & suggest some constructions before proceeding to the step below where the classical construction is provided.
3) Click on the 'Extend DE' and 'Show quadrilateral ADCF' buttons for a construction that should help you with the guided proof in the associated Worksheet: Reasoning Backwards: Triangle Midpoints.
4) Work through the guiding questions given in the accompanying worksheet (see link above) to develop a proof.
5) As asked in the Further Exploration section of the accompanying worksheet above, formulate a converse and click on the 'Link to Triangle Midpoints Converse' button to navigate to a new sketch.
6) Use this new sketch to check whether your formulated converse is true or not by using the given Tools in left margin (scroll down if necessary).
7) Challenge: Can you explain why (prove that) your observations in 5) are true?
Further Generalization
8) What happens if we start with any point on the side of a triangle and draw a line throught that point parallel to a side? Do any interesting results arise?
9) Think about this for a moment and form a conjecture before navigating to this sketch Reasoning Backwards: Parallel Lines to explore these questions further.
Working Backwards (Analysis)
The Reasoning Backwards technique used above in my Rethinking Proof book to illustrate the systematization function of proof is closely connected to the method of Working Backwards or Analysis as it was famously called by Pappus (approx. 290-350 AD) of Alexandria in North Africa. He described the method of working backwards, which he called analysis or solution backwards (Greek: anapalin lysis), as a process where one assumes the desired solution is true and traces its consequences to a known or accepted truth. By then reversing this logical chain, known as synthesis, one can construct the solution from known principles. This "working backwards" technique was a crucial part of the Ancient Greek approach to problem-solving in geometry.
Basically this is how he described the technique:
1. Assume the Goal:
Start by assuming that the problem's solution is true and has been achieved.
2. Trace Consequences:
Follow the chain of implications and consequences, as if they were already established by your initial hypothesis.
3. Reach a Known Truth:
Continue this process until you arrive at a statement that is already known to be true or is a first principle.
4. Reverse the Steps (Synthesis):
Once you find this known truth, reverse the logical order of the steps you took. This reverse process, called synthesis, constructs the actual solution to the original problem.
Note that the last Synthesis step is what Freudenthal refers to above as the anti-didactical inversion, since this is not only how the result is published, but also how it is almost invariable taught to students. This then unfortunately hides from the student how the actual solution was obtained - it seems to the student as if it has appeared brilliantly, almost by magic, whereas the analysis process is actually quite simple & straight-forward, and the solution then becomes naturally apparent.
Read more about the Working Backwards technique of Pappus as described and illustrated in papers by Hee-Chan Lew (2004 & 2006) from the Korea National University of Education.
Unfortunately, the method of synthesis has been in place for more than 2000 years, and is virtually universally entrenched at university, and also at school (even though over the years some teaching experiments at school & university have shown that mathematics can be taught differently). What has also happened in more recent times is that some international mathematics educators have basically taken the approach to still stick to the axiomatic-deductive structure & follow the same sequence, but try & embed & supplement the activities with some experimentation, conjecturing, disproving, etc. using dynamic geometry, or even symbolic mathematic processing systems like Mathematica. While this is an improvement on the traditional approach, this is still a far cry from giving students a more authentic view of how mathematics develops, and how it is often restructured and re-organized several times over. Clearly it is extremely difficult (perhaps even impossible?) to change the predominant two thousands years old status quo of the axiomatic-deductive (Euclidean) structuring of learning materials & their teaching. In addition, there are also other entrenched reasons for this status quo dominance (see De Villiers (1978; 1986) for more details).
References
De Villiers, M. (1978, 2011). Boolean Algebra at School, Vol. 1 (free to download). Dynamic Mathematics Learning, Lulu Press.
De Villiers, M. (1978, 2011). Boolean Algebra at School, Vol. 2 (Teacher Notes) (free to download). Dynamic Mathematics Learning, Lulu Press.
De Villiers, M. (1986). The Role of Axiomatisation in Mathematics and Mathematics Teaching (free to download). Research Report, Research Unit for Mathematics Education (RUMEUS), University of Stellenbosch, South Africa.
De Villiers, M. (1987). Teaching Modeling and Axiomatization with Boolean Algebra (free to download). Mathematics Teacher, Oct, pp. 528-532.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
Freudenthal, H. (1973). Mathematics as an Educational Task. Dordrect, D. Reidel Publishing Company.
Lew, H-C. (2004). The “analysis” of construction problems in the dynamic geometry. Paper delivered at The 9th ATCM Annual meeting, Dec. 15, 2004, Singapore.
Lew, H-C. (2006). Pappus in a Modern Dynamic Geometry: An Honest Way for Deductive Proof. In Hoyles, C; Jean-Baptiste Lagrange, J-B; Son, L.H. & Sinclair, N. (2006). Proceedings of the Seventeenth ICMI Study Conference "Technology Revisited". Held at Hanoi University of Technology, Vietnam; Dec. 3-8 2006, pp. 354-361
Other Rethinking Proof Activities
Other Rethinking Proof Activities
Related Links
Parallel Lines (Rethinking Proof activity)
Reasoning Backwards: Parallel Lines (Rethinking Proof activity)
Kite Midpoints (Rethinking Proof activity)
Isosceles Trapezoid Midpoints (Rethinking Proof activity)
Light Ray in a Triangle (Rethinking Proof activity)
Introducing, Classifying, Exploring, Constructing & Defining Quadrilaterals
Perimeter inscribed parallel-hexagon
Area Ratios (Rethinking Proof activity)
Area ratios of some parallel-polygons inscribed in quadrilaterals and triangles
Merry Go Round the Triangle
(The link above explores a further generalization of Thomsen's hexagon in the related link 'parallel Lines' at the top)
International Mathematical Talent Search (IMTS) Problem Generalized
A Geometric Paradox Explained (Another variation of an IMTS problem)
Crossed Quadrilateral Properties
Midpoint Trapezium Theorem
Midpoint trapezium (trapezoid) theorem generalized
Easy Hexagon Explorations
External Links
Hans Freudenthal (Wikiquote)
Hans Freudenthal: a mathematician on didactics and curriculum theory (PDF)
Pappus of Alexandria (MacTutor)
AMESA - The Association for Mathematics Education of South Africa
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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Michael de Villiers, created with WebSketchpad, 18 Sept 2025.