Problem: Construct all conics tangent to four given lines determined by the sides of a quadrilateral q = A_{1}A_{2}A_{3}A_{4} (may be convex or not).

Analysis: A first crucial fact is that the diagonals and the lines joining opposite tangency points X_{1}X_{2} and Y_{1}Y_{2} intersect at the same point C_{3}. C_{1}, C_{2}, C_{3} are the intersection points of opposite sides of q. A second crucial fact is that C_{3} is the pole of the line C_{1}C_{2} w.r. to every conic inscribed in q. A third crucial fact is that X, Y are harmonic conjugate w.r. to B_{13}, B_{24}. These three facts suffice to complete a recipe for the construction of all conics tangent to the sides of quadrilateral q. Construction: Start with an arbitrary point X on line C_{1}C_{2}. Construct the conjugate harmonic Y of X w.r. to B_{13}, B_{24}. Join X, Y with C_{3} and define points X_{1}, X_{2}, Y_{1}, Y_{2} through the intersections with the sides of q. Join Y_{1} with C_{1} and define the intersection point W with line X_{1}X_{2}. Define the harmonic conjugate Z of Y_{1} w.r. to C_{1} and W (on line C_{1}Y_{1}). The five points {X_{1},X_{2},Y_{1},Y_{2}, Z} are on a conic tangent to the sides of q. Remark Note that the centers O of all these conics lie on the Newton line of the quadrilateral i.e. the line O_{1}O_{2} joining the middles of the diagonals (see Newton.html ). Note also that this line is parallel to the symmetry axis of the unique parabola tangent to the sides of q. In that case O is the point at infinity of line O_{1}O_{2}. For an analytic proof of the remark see: Aubert, P & Papelier, G. Exercices de Geometrie Analytique Paris, Librairie Vuibert, 1966, p. 249.