harmonic division

• If the point $X$ is on the line segment $AB$ and  $XA:XB=p:q$,  then $X$ divides $AB$ internally in the ratio $p:q$.

• If the point $Y$ is on the extension of line segment $AB$ and  $YA:YB=p:q$,  then $Y$ divides $AB$ externally in the ratio $p:q$.

• If $p:q$ is the same in both cases, then the points $X$ and $Y$ divide $AB$ harmonically in the ratio $p:q$.

Theorem 1. The bisectors of an angle of a triangle and its linear pair divide the opposite side of the triangle harmonically in the ratio of the adjacent sides.

Theorem 2. If the points $X$ and $Y$ divide the line segment $AB$ harmonically in the ratio $p:q$, then the circle with diameter the segment $XY$ (the so-called Apollonius’ circle) is the locus of such points whose distances from $A$ and $B$ have the ratio $p:q$.

The latter theorem may be proved by using analytic geometry.