## 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 Δ*ABC* and its incircle. Label the points where the circle touches the sides *BC*, *CA*, and *AB* respectively as *A*_{1}, *B*_{1} and *C*_{1}. Repeat the process with the new Δ*A*_{1}B_{1}C_{1}. Then again, and again, etc. What do you visually notice about the shape, and the displayed values of the angles of Δ*A*_{n}B_{n}C_{n} as *n* increases? Check by dragging vertices *A*, *B* or *C*. Can you make a conjecture?

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Tangent points of incircle

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

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**2) Excentres**: Start with any Δ*ABC* and construct its *excentres*. Label the excentres formed on the sides of *BC*, *CA*, and *AB* respectively as *A*_{1}, *B*_{1} and *C*_{1}. Repeat the process with the new Δ*A*_{1}B_{1}C_{1}. Then again, and again, etc. What do you visually notice about the shape, and the displayed values of the angles of Δ*A*_{n}B_{n}C_{n} as *n* increases? Check by dragging vertices *A*, *B* or *C*. Can you make a conjecture?

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Excentres

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

Read my paper Over and over again: two geometric iterations with triangles. *Learning & Teaching Mathematics*, July 2014, No. 16, pp. 40-45, 426 KB, PDF.

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Created by Michael de Villiers, 12 March 2014.