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Investigation 1: Tangent points of incircle
Start with any ΔABC and its incircle and incentre I. Label the points where the circle touches the sides BC, CA and AB respectively as A1, B1 and C1 as shown below. Repeat the process with the new ΔA1B1C1 and determine its incentre I1. Then repeat the process twice more. Connect I to I3 with a straight line.
What do you visually notice about the four incentres I, I1, I2 and I3? (The labels for I2 and I3 are not shown to avoid clutter). Check by dragging the red vertices A, B or C. Can you make a conjecture? Can you prove or disprove your conjecture?
Iteration of tangent points of incircle
Investigation 2: Excentres
Start with any ΔABC and its incentre I and excentres. Label the excentres formed on the sides BC, CA and AB respectively as A1, B1 and C1, and construct its incentre I1 as shown below. Repeat the process with the new ΔA1B1C1. Then repeat the process twice more. Connect I to I3 with a straight line.
What do you visually notice about the four incentres I, I1, I2 and I3? Check by dragging the red vertices A, B or C. Can you make a conjecture? Can you prove or disprove your conjecture?
Iteration of excentres
Visit the Sine of the Times blog to access an interactive blog Refutation in a Dynamic Geometry Context and/or read my paper Conjecturing, refuting and proving within the context of dynamic geometry.
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By Michael de Villiers. Created, 18 May 2014.