If are on a line (in that order), then these statements are pairwise equivalent:

- is a harmonic division
- and are inverses w.r.t. the circle with diameter
- and are inverses w.r.t. the circle with diameter
- the circle with diameter is a circle of Apollonius with foci
- the circle with diameter is a circle of Apollonius with foci

If is a point outside :

is right iff is bisected (internally/externally) by and/or ;
is right iff is bisected (internally/externally) by and/or ;


iff

But and so the equation becomes


depending on only the angles between , , , .
This is why we can say is a harmonic pencil.

Suppose is a triangle and the tangents through , to the circumcircle of intersect at . Then is the -symmedian of .

Proof: Let the perpendicular bisector of (also passing through the center of the circumcircle and through ) intersect the circle at , with closer to ; let the midpoint of be .

Then since and are inverses w.r.t. the circumcircle, is a harmonic division. Since is right because is a diameter of the circumcircle, is bisected by . Since is the midpoint of arc , also bisects . Thus and are isogonals w.r.t. . Since is the -median, is the -symmedian.