Droz-Farny Theorem


If two perpendicular lines are drawn through the orthocenter H of any triangle, these lines intercept each side (or its extension) in two points (labeled P_(12), P_(12)^', P_(13), P_(13)^', P_(23), P_(23)^'). Then the midpoints M_(12), M_(12), and M_(23) of these three segments are collinear.

The two given lines, the lines connecting the midpoints and the sides of the reference triangle are all tangent to the same (inscribed) parabola. Instead of the midpoints, one may take any other ratio t with


and the points M_(23), M_(13), and M_(12) will still be collinear in addition to bing tangent to the same parabola (Ehrmann and van Lamoen 2004).

See also

Collinear, Midpoint

Portions of this entry contributed by Floor van Lamoen

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Ayme, J.-L. "A Purely Synthetic Proof of the Droz-Farny Line Theorem." Forum Geom. 4, 219-224, 2004., A. "Droz-Farny Line Theorem.", A. "Question 14111." Ed. Times 71, 89-90, 1899.Ehrmann, J.-P. and van Lamoen, F. M. "A Projective Generalization of the Droz-Farny Line Theorem." Forum Geom. 4, 225-227, 2004., R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 73, 1995.Sharygin, I. Problem II 206 in Problemas de Geometria. Moscow: Mir, pp. 111 and 311-313, 1986.Thas, C. "A Note on the Droz-Farny Theorem." Forum Geom. 6, 25-28, 2006.

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Droz-Farny Theorem

Cite this as:

van Lamoen, Floor and Weisstein, Eric W. "Droz-Farny Theorem." From MathWorld--A Wolfram Web Resource.

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