Investigation: Circumcentres of iterated median triangles
1) Start with any ΔABC and construct its circumcentre O as shown below. Label the midpoints of 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 circumcentre O1. Then repeat the process once more. Connect O to O1 with a straight line.
2) What do you visually notice about the three circumcentres O, O1, O2 and the centroid G? Check by dragging the red vertices A, B or C.
3) Can you make a conjecture? Can you prove or disprove your conjecture?
4) Click on the 'Show Measurements' button. What do you notice? Can you prove or disprove these properties as well? How do they relate to your conjecture in 3)?
Investigating circumcentres of iterated median triangles
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Reference
De Villiers, M. & Heideman, N. (2014). Conjecturing, refuting and proving within the context of dynamic geometry. Learning & Teaching Mathematics, no. 17, pp. 20-26.
Explore More
5) Explore on your own using dynamic geometry what happens if instead of circumcentres of the iterated median triangles, you instead construct incentres or orthocenters?
(Alternatively, click on the 'Link to' buttons to navigate to sketches displaying the incentres or orthocentres of the iterated median triangles).
6) What do you anticipate will happen if instead of circumcentres of the iterated median triangles, you instead construct their centroids?
7) Can you prove your observations in 5) and 6) above? Can you generalize further?
(Proof Hint: For all three cases above, have a look at this interactive, dynamic sketch: Euler line proof.)
Related Links
Investigating incentres of some iterated triangles
Over and over: two geometric iterations with triangles
Area Ratios (Rethinking Proof activity)
Concurrency Conjecture (Rethinking Proof activity)
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
Point Mass Centroid (centre of gravity or balancing point) of Quadrilateral
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
Four Collinear Points
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Michael de Villiers, created 18 May 2014 with JavaSketchpad; updated to WebSketchpad, 12 July 2025; 12 June 2026.