The proposition relating to the bisector of an exterior angle may be considered as a part of this one, but it is usually treated separately in order that the proof shall appear less involved, although the two are discussed together at this time. The proposition relating to the exterior angle was recognized by Pappus of Alexandria.
If ABC is the given triangle, and CP_1, CP_2 are respectively the internal and external bisectors, then AB is divided harmonically by P1 and P2.
∴AP1 : P1B = AP2 : P2B.
∴AP2 : P2B = AP2 - P1P2 : P1P2 - P2B,
and this is the criterion for the harmonic progression still seen in many algebras. For, letting AP2 = a, P1P2 = b, P2B = c, we have
which is also derived from taking the reciprocals of a, b, c, and placing them in an arithmetical progression, thus:
1/b - 1/a = 1/c - 1/b,
whence (a - b)/ab = (b - c)/bc,
or (a - b)/(b - c) = ab/bc = a/c.
This is the reason why the line AB is said to be divided harmonically. The line P1P2 is also called the harmonic mean between AP2 and P2B, and the points A, P1, B, P2 are said to form an harmonic range.
It may be noted that ∠P2CP1, being made up of halves of two supplementary angles, is a right angle. Furthermore, if the ratio CA : CB is given, and AB is given, then P1 and P2 are both fixed. Hence C must lie on a semicircle with P1P2 as a diameter, and therefore the locus of a point such that its distances from two given points are in a given ratio is a circle. This fact, Pappus tells us, was known to Apollonius.