This makes a pretty picture. Drag the slider slowly to the left to reveal each line or point one at a time. If you do anything with conics and/or their constructions, this is an unexpected conic, which are the best kind!
Draw the Pedal Line. Recall the Pedal Line is the line through the feet of the perpendiculars dropped from the Focal Point to each side. Recall the Focal Point is the intersection point of the four circumcircles of the Four Triangles.
Draw lines through the midpoints of pairs of opposite sides.
These seven lines, along with the four lines of the quadrilateral, are tangent to the parabola whose focus is the Focal Point and whose directix is the Orthocentric Line.