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 ΔA_nB_nC_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 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 ΔA_nB_nC_n as n increases? Check by dragging vertices A, B or C. Can you make a conjecture?

Please install Java (version 1.4 or later) to use JavaSketchpad applets.

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.