Given two pairs of points (A,B) and (C,D) on the same line (e) there is another pair (X,Y) which is simultaneously harmonic conjugate to (A,B) and (C,D). The pair (X,Y) is real only in the case in which (A,B) does not separates (C,D) or, equivalently, the circle bundle (I) generated by the circles {cAB,cCD} with diameters correspondingly {AB, CD} is of non intersecting type. The pair (X,Y) then coincides with the limit-points of this bundle. Consequently the (X,Y) are the intersection points with (e) of a circle (c) intersecting both circles orthogonally (i.e. belonging to the orthogonal bundle (II) of (I)).
![[alogo]](alogo.png){kind=small}
Common harmonic (X,Y) to two pairs (A,B) and (C,D) ![[0_0]](HarmonicCommon_files/HarmonicCommon_0_0_0.png)
![[0_1]](HarmonicCommon_files/HarmonicCommon_0_0_1.png)