Consider two intersecting circles at points A,B. Draw a line L through A and measure the length x of segment AEF, defined by the two other intersection points {E,F} of L with the circles. Length x obtains the maximum value when EF is orthogonal to AB, i.e. {E,F} coincide with the diametral points {D,C} of B.

A simple consequence of the similarity of triangles BDC and BEF and the fact that BDC is maximal, since diameters are the maximal chords of a circle. Remark Note that the maximal value of EF is the double of the diacentric distance |OP| of the two circles.