Cyclic Quadrilateral - Anticenter and Collinearities

Anticenter and Collinearities

Four line segments, each perpendicular to one side of a cyclic quadrilateral and passing through the opposite side's midpoint, are concurrent. These line segments are called the maltitudes, which is an abbreviation for midpoint altitude. Their common point is called the anticenter. It has the property of being the reflection of the circumcenter in the "vertex centroid". Thus in a cyclic quadrilateral, the circumcenter, the "vertex centroid", and the anticenter are collinear.

If the diagonals of a cyclic quadrilateral intersect at P, and the midpoints of the diagonals are M and N, then the anticenter of the quadrilateral is the orthocenter of triangle MNP. Moreover, the anticenter is the midpoint of the line segment joining the midpoints of the diagonals.

In a cyclic quadrilateral, the "area centroid" G_a, the "vertex centroid" G_v, and the intersection P of the diagonals are collinear. The distances between these points satisfy