Circumcircle of Pedal Triangle of an Orthic Quadruple
- Warren Koepp
In this construction A, B, C are free points. Any three of A, B, C, D form a triangle for which the fourth point is the orthocenter (i.e. point of concurrency of the three altitudes). This could be any of the four possible triangles with "thickened" segments as edges; red is used to focus on triangle ABC. For any of the triangles with all three vertices among A, B, C, D, we call triangle EFJ (formed by the 3 altitude feet) its 'pedal triangle'.
Think through the various triangles and the claims and properties in the text box, until they make sense. Reposition A, B, C to form various triangles, and observe how the altitude feet and segment midpoints move, but always remain on the circumcircle of triangle EFJ. This circle is the 'nine point circle' of any of the four triangles having as vertices three of the points A, B, C, D. The midpoint of a segment connecting a vertex to the orthocenter (the three black segments if we focus on triangle ABC) is called an "Euler point" of the triangle. Depending on which triangle is your focus, three of the segment midpoints are midpoints of edges, and the other three are Euler points.