Drag the vertices A, B, and C to adjust the triangle.
The obscure 19th century geometer Karl Feuerbach discovered one of the most beautiful objects in elememtary geometry.
He studied the circumscribed circle (shown in green) of the triangle formed by the feet (labeled F[A], F[B], and F[C]) of the altitudes of a triangle ABC.
As the diagram suggests, the midpoints (labeled AB/2, BC/2, and CA/2) of the sides of the original triangle also lie on this circle, as do the midpoints (labeled OA/2, OB/2, and OC/2) of the segments of the altitudes joining the orthocenter to the vertices! This property (and Feuerbach's obscurity) give the circle its modern name: the 9 Point Circle.
The 9 point circle has a remarkable relationship to the Euler segment. Feuerbach was interested in the even more remarkable relationship to the inscribed circle.
While it's not exactly as simple as 1-2-3, I am especially fond of the way the feet of the altitudes split up the other 6 points.
If you have Geometer's Sketchpad, you might enjoy the sketch upon which this page is based.
Copyright © 2003 Joe Christy joe @ eshu.net.
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