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1) Given any right triangle ABC with incircle with centre I, and excircles with centres D, E and F as shown.
2) Drag any of the vertices of right triangle ABC, and observe the measurements of the radii, and the given calculation. What do you notice?
Conjecture: Formulate a conjecture, and check by dragging any of A, B or C.
SA Mathematics Olympiad Problem 2016, Round 2, 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 2, Question 20, where the radii for circles I, D and E were given and learners were asked to determine the radius of excircle F on the hypotenuse. The 2016 Question Paper, Round 2 is available at here.
Some Different Solutions:
Read my paper in Learning & Teaching Mathematics, no. 22, July 2017 at "A Multiple Solution Task: Another SA Mathematics Olympiad Problem".
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Created by Michael de Villiers, 18 May 2016.