Advanced High-School Mathematics

SECTION 1.4 Harmonic Ratio 41

set A;X;B =

AX

XB

(signed magnitudes); if A;X;B > 0 we call this

quantity the internal divisionof [AB], and if A;X;B < 0 we call
this quantity theexternal divisionof [AB]. Finally, we say that the
colinear pointsA, B, X, andY are in aharmonic ratioif

A;X;B = −A;Y;B;

that is to say, when

AX

XB

= −

AY

Y B

(signed magnitudes).

It follows immediately from the Angle Bisector Theorem (see page 15)
that when (BX) bisects the interior angle atCin the figure above and
(BY) bisects the exterior angle atC, thenA, B, X,andY are in har-
monic ratio.

Note that in order for the pointsA, B, X, andY be in a harmonic
ratio it is necessary that one of the pointsX, Y be interior to [AB] and
the other be exterior to [AB]. Thus, ifX is interior to [AB] andY is
exterior to [AB] we see that A, B, X, andY are in a harmonic ratio
precisely when

Internal division of [AB] by X = −External division of [AB] byY.

Exercises

1. LetA,B,andC be colinear points with (A;B;C)(B;A;C) =−1.
   Show that thegolden ratiois the positive factor on the left-hand
   side of the above equation.


2. Let A, B, and C be colinear points and let λ = A;B;C. Show
   that under the 6=3! permutations ofA, B, C, the possible values
   ofA;B;C are

λ,

1

λ

,−(1 +λ),−

1

1 +λ

,−

1 +λ

λ

,−

λ

1 +λ

.