The Second Lemoine Circle
Through the symmedian point K of a triangle ABC draw lines parallel to the sides of the orthic triangle. The six points of intersection of those lines with the sides of the triangle lie on a circle (the second Lemoine circle) with center K.
The triangle formed by the intersection of the altitudes with the sides of a triangle ABC is called the orthic triangle of ABC.
The centroid of a triangle is the intersection of the lines drawn from the vertices to the midpoints of the opposite sides.
Let P be a point inside ABC. The reflections of the three lines AP, BP, and CP in the angle bisectors at A, B, and C meet in a point, called the isogonal conjugate of P.
The symmedian point is the isogonal conjugate of a triangle's centroid.