Christopher J Bradley and Geoff C. Smith, On a construction
Forum Geometricorum, 7 (2007) 231--247.
Abstract: In 1907 Hagge constructed a circle associated with each Cevian
point P of triangle ABC. If P is on the circumcircle this circle degenerates
to a straight line through the orthocenter which is parallel to the Wallace-Simson
line of P. We give a new proof of Hagge's result by a method based on re ections.
We introduce an axis associated with the construction, and (via an areal
analysis) a conic which generalizes the nine-pont circle. The precise locus
of the orthocenter in a Brocard porism is identifed by using Hagge's theorem
as a tool. Other natural loci associated with Hagge's construction are discussed.