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: Reasoning Backwards: Parallel Lines.
Reasoning Backwards: Parallel Lines
In the earlier Parallel Lines activity, we used the result that a line parallel to one side of a triangle divides the other two sides in the same ratio.
There we used it as an assumption without any proof.
But why is it true that a line parallel to one side of a triangle divides the other two sides in the same ratio? Can we also prove it?
The activity below, and associated worksheet (see link above), deals with these questions.
Reasoning Backwards: Parallel Lines
Notes
1) Freudenthal (1973) argued that that we should as far as possible try to use a teaching approach of re-invention in the teaching of mathematics, and not simply give children ready-made, pre-packaged, already ordered, systematized mathematics. In his view children should first engage with deductive proof in the context of local axiomatization before progressing to global axiomatization. The 1923 & 1939 Reports of the Mathematical Association on The Teaching of Geometry in Schools also recommended at least 3 stages for teaching geometry, namely:
A. An experimental phase which concentrates on the solution of real life problems (e.g. land surveying) and constructions, and which provide a context in which the underlying results regarding angles at a point, parallel lines and congruent triangles are introduced. Deduction can be done if the opportunity presents itself, but formal written proofs should be left for the next phase.
B. A deductive phase in which proof is formally introduced in regard to interesting results which are not self-evident to pupils.
C. A systematization phase in which proof is utilized to deductively order familiar results, and to deduce them from a small number of independent axioms.
In agreement with Stage B, in the earlier Parallel Lines activity, the result that a line parallel to one side of a triangle divides the other two sides in the same ratio was assumed without proof in order to prove the Parallel Lines result (which students usually find surprising). However, in the activity above, the focus shifts more globally to Stage C and to systematizing this earlier assumption and proving it - see Human et al (1987) & De Villiers (1986). A similar reasoning backwards approach was used in an experimental course on Boolean Algebra to arrive at its axioms (De Villiers, 1978).
2) As asked in the accompanying worksheet above, formulate a converse and click on the 'Link to Converse' button to navigate to a new sketch.
3) Use this new sketch to check whether the converse is true or not.
4) In the new sketch use the Dilate tool on the left to first select the ratio AD/AB as the scale factor, then select A as the centre of the dilation and finally select C to apply the dilation to it.
(Note that this dilation has now constructed a point C' (label not yet displayed) so that AC'/AC = AD/AB, which in turn implies that AD/DB = AC'/C'C.)
5) Next use the Segment tool on the left (scroll down if necessary) to construct a line segment connecting D with the newly constructed point C' in 4).
6) Now use the Slope tool to measure the slope of the line segment DC' in 5).
7) What do you notice about the slopes of DC' and BC? Check by dragging.
8) Challenge: Can you explain why (prove that) your observations in 6) & 7) are true?
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. (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.
Human, P.G. & Nel, J. H. et al. (1987). Alternative Instructional Strategies for Geometry Education: A Theoretical and Empirical Study (free to download). (Final theoretical part of the report of the University of Stellenbosch Experiment in Mathematics Education (USEME)-project: 1977-78), University of Stellenbosch.
Mathematical Association (1923, 2nd edition 1925). The Teaching of Geometry in Schools (free to download). London: Bell & Sons.
Mathematical Association (1939, 1954 reprint). A Second Report on The Teaching of Geometry in Schools (free to download). London: Bell & Sons.
Other Rethinking Proof Activities
Other Rethinking Proof Activities
Related Links
Parallel Lines (Rethinking Proof activity)
Kite Midpoints (Rethinking Proof activity)
Isosceles Trapezoid Midpoints (Rethinking Proof activity)
Light Ray in a Triangle (Rethinking Proof activity)
Reasoning Backward: Triangle Midpoints (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 dynamic sketch 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
Intercept theorem (Wikipedia)
Euclid's Elements: Book VI, Proposition 2
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, 26 July 2025; updated 27 July 2025.