If O is the circumcentre of triangle ABC with K, L and M the respective midpoints of BC, CA and AB, and points D, E and F are respectively constructed on the lines OA, OB and OC so that OD = OE = OF, then the lines DK, EL and FM are concurrent in X and the locus of X lies on the Euler line.
Explore
To view the locus, drag D to make the circle bigger or smaller.
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Concurrency and Euler line locus result
Challenge
Can you explain (prove) why the result is true?
If not, click on Solution.
Related Links
Euler line proof
Nine Point Conic and Generalization of Euler Line
Further Euler line generalization
Six Point Cevian Circle & Conic
Nine-point centre (anticentre or Euler centre) & Maltitudes of Cyclic Quadrilateral
The quasi-Euler line of a quadrilateral and a hexagon
A side trisection triangle concurrency
External Links
Nine-point circle
Euler line
The Euler Line and the 9-Point Circle
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Michael de Villiers, 17 May 2010; updated to WebSketchpad, 20 March 2024.