- Cosmin Pohoata, Son Hong Ta, A short proof of Lamoen’s generalization of the Droz-Farny line theorem, preprint. (Link to the PDF file)
In 1899, Arnold Droz-Farny discovered the following beautiful result, known nowadays as the Droz-Farny line theorem: If two perpendicular straight lines are drawn through the orthocenter of a triangle, they intercept a segment on each of the sidelines. The midpoints of these three segments are collinear. Lamoen’s slightly more general version says that if the midpoints of the intercepted segments are replaced by three points dividing into the same ratio the corresponding segments, then these new points remain collinear. We give a short “halfway” synthetic proof of this fact.
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