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.

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