SA Mathematics Olympiad Problem 2016, Round 2, Question 20

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Exploration: 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

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 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.