consider the parabola tangent to all four sides of the quadrilateral. by the reflection property of a parabola, we get that the focus of this parabola must be the Miquel point by simson line theorem. however a parabola is just an inellipse with one foci at infinity as well. so the other focus of the parabola is a point at infinity, and therefore the isogonal conjugate of the miquel point is a point at infinity. this implies that its reflections over all four sides must be collinear (since the circumcenter of a point's reflections over all four sides is its isogonal conjugate). however this is also the steiner line of the miquel point in all of the triangles !! so all four orthocenters are collinear