This interactive, dynamic sketch illustrates a beautiful 'duality' (or analogy) between the famous Nine-point circle and the Spieker circle, as well as a corresponding duality between the Euler line and the Nagel line.
In this 'duality' the concepts of inscribed and circumscribed circles of the median triangle are interchanged as well as the concepts of incentre and circumcentre, and also the concepts of orthocentre and Nagel point.
The observation of this 'duality' is not new, and appears, for example, together with proofs in Coolidge (1st published 1916) and Honsberger (1995).
Coolidge, J.L. (1971 Edition). A Treatise on the Circle and the Sphere (pp. 53-57). Bronx, NY: Chelsea Publishing Company (original publication 1916).
Honsberger, R. (1995). Episodes in Nineteenth & Twentieth Century Euclidean Geometry (pp. 7-13). The Mathematical Association of America. Washington: MAA.
Nine Point Circle and Spieker Circle Duality
1) Drag points A, B, or C in triangles ABC.
2) Dynamically explore a further generalization of the nine-point circle to a Nine Point Conic and generalization of the Euler line.
3) Dynamically explore a further generalization of the Spieker circle to a Spieker Conic and generalization of Nagel line. This generalization utilized the 'duality' illustrated above.
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Michael de Villiers, 14 June 2017 using WebSketchpad.