NOTE: Please WAIT while the applet below loads.
1) Start with any square ABCD and an arbitrary point E on AD (extended). Bisect ∠EBC with ray BF, and F on CD (extended).
2) Drag point E, and observe the measurements AE (a), CF and BE. Click on the 'Show Measurement' button. What do you notice?
Conjecture: Formulate a conjecture, and check by dragging E along AD (also on to its extensions).
SA Mathematics Olympiad Problem 2016, Round 1, Question 20
Can you explain why (prove) your conjecture above is true? If so, can you find other, different ways of logically explaining (proving) your conjecture?
A special case of this theorem was used for the Senior SA Mathematics Olympiad 2016, Round 1, Question 20, where AE = 2 and CF = 3 was given and learners were asked to determine BE. The 2016 Question Paper, Round 1 is available at here.
If you get stuck, or would like to compare your solution with some other possible solutions, read my paper in the June 2016 issue of Learning & Teaching Mathematics paper: A Multiple Solution Task: a SA Mathematics Olympiad Problem.
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Created by Michael de Villiers, 26 April 2016.