Investigation 1: Tangent points of incircle
1) 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.
2) 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?
Investigating incentres of some iterated triangles
Investigation 2: Excentres
3) Click on the 'Link to Investigation 2' button to navigate to a new sketch.
4) Start with any ΔABC and its incentre I and its three 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.
5) 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?
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Visit the Sine of the Times blog to access an interactive blog Refutation in a Dynamic Geometry Context and/or read my joint 2014 paper with Nic Heideman in the Learning & Teaching Mathematics journal, no. 17, pp. 20-26, namely, Conjecturing, refuting and proving within the context of dynamic geometry.
Related Links
Investigating circumcentres of iterated median triangles
Over and over: two geometric iterations with triangles
False Collinear Conjecture
Isosceles Triangle Collinear Conjecture
Triangle Incentre-Circumcentre Collinearity
Some Trapezoid (Trapezium) Explorations (see Investigations 3 & 4)
An interesting collinearity
Euler line proof
Further Euler line generalization
Euler-Nagel line analogy
Nine Point Conic and Generalization of Euler Line
Spieker Conic and generalization of Nagel line
More Properties of a Bisect-diagonal Quadrilateral
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
Concurrency, collinearity and other properties of a particular hexagon
Quadrilateral Similar Triangles Collinearity
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Michael de Villiers, created 18 May 2014 with JavaSketchpad; updated to WebSketchpad, 11 July 2025.