The Nine-Point Conic

Three cevians are concurrent at point P lead to nine points. Three are formed by the cevians intersecting the sides at points E, F, and D. The midpoints from point P to each vertex are L, I, and J. The midpoints of the sides are M, K, and N. These nine points lie on a conic. When P is the orthocenter H, the conic is the nine-point circle, hence, a special case of the nine-point conic. When P is the centroid G, the conic is the Steiner inellipse, the unique ellipse tangent to the side of the triangle at the midpoints of the sides. When P is outside the triangle, the conic is a hyperbola. When P is on the circumcircle, the conic passes through the circumcenter O.