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.

Suppose that the vertices of the quadrilateral

Let

be the perpendicular bisectors of sides

Then their intersections

, with subscripts considered modulo 4, form the consequent quadrilateral

The construction is then iterated on

to produce

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

is not degenerate.

[1] Combining #1 and #2,

is always nondegenrate.

are homothetic, and in particular, similar.

are also homothetic.

The perpendicular bisector construction can be reversed via isogonal conjugation.

, it is possible to construct

Let

α , β , γ , δ

be the angles of

, the ratio of areas of

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

is tangential then

is also tangential.