page_title: Perpendicular bisector construction of a quadrilateral
section_title: Definition of the construction
hierarchical_section_title: Perpendicular bisector construction of a quadrilateral / Definition of the construction
context_page_description: 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.
context_section_description: 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 .
caption: First iteration of the perpendicular bisector construction