Perpendicular bisector construction of a quadrilateral

In geometry, the perpendicular bisector construction of a quadrilateral is a construction which produces a new quadrilateral from a given quadrilateral using the perpendicular bisectors to the sides of the former quadrilateral. This construction arises naturally in an attempt to find a replacement for the circumcenter of a quadrilateral in the case that is non-cyclic.

Definition of the construction
Suppose that the vertices of the quadrilateral $$ Q $$ are given by $$ Q_1,Q_2,Q_3,Q_4 $$. Let $$ b_1,b_2,b_3,b_4 $$ be the perpendicular bisectors of sides $$ Q_1Q_2,Q_2Q_3,Q_3Q_4,Q_4Q_1 $$ respectively. Then their intersections $$ Q_i^{(2)}=b_{i+2}b_{i+3} $$, with subscripts considered modulo 4, form the consequent quadrilateral $$ Q^{(2)} $$. The construction is then iterated on $$ Q^{(2)} $$ to produce $$ Q^{(3)} $$ and so on.

An equivalent construction can be obtained by letting the vertices of $$ Q^{(i+1)} $$ be the circumcenters of the 4 triangles formed by selecting combinations of 3 vertices of $$ Q^{(i)} $$.

Properties
1. If $$ Q^{(1)} $$ is not cyclic, then $$ Q^{(2)} $$ is not degenerate.

2. Quadrilateral $$ Q^{(2)} $$ is never cyclic. Combining #1 and #2, $$ Q^{(3)} $$ is always nondegenrate.

3. Quadrilaterals $$ Q^{(1)} $$ and $$ Q^{(3)} $$ are homothetic, and in particular, similar. Quadrilaterals $$ Q^{(2)} $$ and $$ Q^{(4)} $$ are also homothetic.

3. The perpendicular bisector construction can be reversed via isogonal conjugation. That is, given $$ Q^{(i+1)} $$, it is possible to construct $$ Q^{(i)} $$.

4. Let $$ \alpha, \beta, \gamma, \delta $$ be the angles of $$ Q^{(1)} $$. For every $$ i $$, the ratio of areas of $$ Q^{(i)} $$ and $$ Q^{(i+1)} $$ is given by


 * $$ (1/4)(\cot(\alpha)+\cot(\gamma))(\cot(\beta)+\cot(\delta)). $$

5. If $$ Q^{(1)} $$ is convex then the sequence of quadrilaterals $$ Q^{(1)}, Q^{(2)},\ldots $$ converges to the isoptic point of $$ Q^{(1)} $$, which is also the isoptic point for every $$ Q^{(i)} $$. Similarly, if $$ Q^{(1)} $$ is concave, then the sequence $$ Q^{(1)}, Q^{(0)}, Q^{(-1)},\ldots $$ obtained by reversing the construction converges to the Isoptic Point of the $$ Q^{(i)} $$'s.

6. If $$ Q^{(1)} $$ is tangential then $$ Q^{(2)} $$ is also tangential.