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If perpendicular lines intersecting at the orthocenter are allowed to cut all three sides (extended) of a triangle they will cut each side in two places. The Droz-Farney theorem states that the midpoints of the two intersections on each side of the triangle will lie on a straight line.

In the figure, the perpendiculars are in bold blue. The midpoints of each pair of intersections are labled M1, M2, M3. The red line is the line predicted by the theorem.

The vertices of the triangle can be moved, and the direction of the perpendiculars may be changed at the "move here" point.

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droz-farny