when two circles have a common chord, their centers and the endpoints of the chord form a quadrilateral.

Question

What kind of quadrilateral is it? What special property do its diagonals have?

Answer 1

The centers of the circles are equidistant from the endpoints of the common chord, so the centers determine a perpendicular bisector of the chord. Hence the diagonal joining the centers of the circles bisect the chord (the other diagonal) perpendicularly. This means that the quadrilateral is a kite (or a rhombus if the radii of the circles are congruent).