On the Nine Point Circle

Abstract

ABC is a triangle; A'B'C the middle points of the sides forming the medial triangle; DEF the orthocentric or pedal triangle; S the circumcentre; 0 the orthocentre; A₁, B₁, C₁ the middle points of OA, OB, OC; G the centroid; I the in-centre; N the nine point circle centre; K the symmedian point; a,b,c the points of contact of the inscribed circle with the sides; Ï€ the Feuerbach point for the incircle; d the point of contact with the Steiner ellipse of the common tangent of the Steiner ellipse, inscribed and nine point circles; T the Gergonne point of intersection of Aa, Bb, Cc.