SA Mathematics Olympiad Problem 2016, Round 1, Question 20

NOTE: Please WAIT while the applet below loads.

Exploration: 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

Explanation (proof): Can you explain why (prove) your conjecture above is true? If so, can you find other, different ways of logically explaining (proving) your conjecture?

Remark:
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.