## Over and over: two geometric iterations with triangles

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(For a definition of iteration go here.)

1) Tangent points of incircle: Start with any ΔABC and its incircle. Label the points where the circle touches the sides BC, CA, and AB respectively as A1, B1 and C1. Repeat the process with the new ΔA1B1C1. Then again, and again, etc. What do you visually notice about the shape, and the displayed values of the angles of ΔAnBnCn as n increases? Check by dragging vertices A, B or C. Can you make a conjecture?

Can you explain why (prove that) your conjecture above is true?

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2) Excentres: Start with any ΔABC and construct its excentres. Label the excentres formed on the sides of BC, CA, and AB respectively as A1, B1 and C1. Repeat the process with the new ΔA1B1C1. Then again, and again, etc. What do you visually notice about the shape, and the displayed values of the angles of ΔAnBnCn as n increases? Check by dragging vertices A, B or C. Can you make a conjecture?