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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 socalled 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.
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