A Quadrangle Center

Let P1, P2, P3, P4 be the defining Quadrangle Points.
Let S1 = P1.P4 /\ P2.P3, S2 = P1.P3 /\ P2.P4 and S3 = P1.P2/\ P3.P4.
Now S1 S2 S3 is the QA-Diagonal Triangle of the Reference Quadrangle.

For each vertex Pi, we take the triangle TjTkTl, where Tj the intersection of the sideline PkPl with circumcircle of the triangle S1S2S3 (other than S1, S2, S3)
Qi = Perspector of the triangle PjPkPl and TjTkTl.

The "unknown" Quadrangle Center is the common intersection point of lines PiQi (i=1,2,3,4)

Randy Hutson (Hyacinthos message #21076):
Let Qi be the cyclocevian conjugate of Pi wrt PjPkPl. QiQjQkQl could be called the 'Cyclocevian Quadrangle' of PiPjPkPl, and its perspector is this new center, which might be called the 'Cyclocevian Center'.
This will be the new point QA-P38 (Hyacinthos message #21077).

Another construction for QA-P38 (Hyacinthos message #21078):
In the complete quadriangle determined by the four points P1, P2, P3, P4, the six lines which join them, cut the circumcircle of the diagonal triangle, S1S2S3, into six points (other than S1, S2, S3). The lines joining pairs of these points, which are on opposite sides of the complete quadriangle, are concurrent in the new point "QA-P38"