## Investigating incentres of some iterated triangles

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**Investigation 1: Tangent points of incircle**

Start with any Δ*ABC* and its incircle and incentre *I*. Label the points where the circle touches the sides *BC*, *CA* and *AB* respectively as *A*_{1}, *B*_{1} and *C*_{1} as shown below. Repeat the process with the new Δ*A*_{1}B_{1}C_{1} and determine its incentre *I*_{1}. Then repeat the process twice more. Connect *I* to *I*_{3} with a straight line.

What do you visually notice about the four incentres *I*, *I*_{1}, *I*_{2} and *I*_{3}? (The labels for *I*_{2} and *I*_{3} 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?

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Iteration of tangent points of incircle

**Investigation 2: Excentres**

Start with any Δ*ABC* and its incentre *I* and excentres. Label the excentres formed on the sides *BC*, *CA* and *AB* respectively as *A*_{1}, *B*_{1} and *C*_{1}, and construct its incentre *I*_{1} as shown below. Repeat the process with the new Δ*A*_{1}B_{1}C_{1}. Then repeat the process twice more. Connect *I* to *I*_{3} with a straight line.

What do you visually notice about the four incentres *I*, *I*_{1}, *I*_{2} and *I*_{3}? 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
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