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 are given by . Let be the perpendicular bisectors of sides respectively. Then their intersections , with subscripts considered modulo 4, form the consequent quadrilateral . The construction is then iterated on to produce and so on.
An equivalent construction can be obtained by letting the vertices of be the circumcenters of the 4 triangles formed by selecting combinations of 3 vertices of .
Properties
1. If is not cyclic, then is not degenerate.
2. Quadrilateral is never cyclic. Combining #1 and #2, is always nondegenrate.
3. Quadrilaterals and are homothetic, and in particular, similar. Quadrilaterals and are also homothetic.
3. The perpendicular bisector construction can be reversed via isogonal conjugation. That is, given , it is possible to construct .
4. Let be the angles of . For every , the ratio of areas of and is given by
5. If is convex then the sequence of quadrilaterals converges to the isoptic point of , which is also the isoptic point for every . Similarly, if is concave, then the sequence obtained by reversing the construction converges to the Isoptic Point of the 's.
6. If is tangential then is also tangential.