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.
![[alogo]](alogo.png){kind=small}
Maximal segment ![[0_0]](MaximalSegment_files/MaximalSegment_0_0_0.png)
![[0_1]](MaximalSegment_files/MaximalSegment_0_0_1.png)