On the Conic Through the Intercepts of the Three Lines Through the Centroid and the Intercepts of a Given Line

Let L be a line intersecting the sidelines of triangle ABC at X , Y , Z respectively. The lines joining these intercepts to the centroid give rise to six more intercepts on the sidelines which lie on a conic Q(L , G). We show that this conic (i) degenerates in a pair of lines if L is tangent to the Steiner inellipse, (ii) is a parabola if L is tangent to the ellipse containing the trisection points of the sides, (iii) is a rectangular hyperbola if L is tangent to a circle CG with center G. We give a ruler and compass construction of the circle CG. Finally, we also construct the two lines each with the property that the conic Q(L , G) is a circle.