Geometry • Topic 32

Harmonic Division

Harmonic division is a special configuration of four points where the cross ratio is $- 1$ . It appears naturally in angle bisectors and polar/pole relationships.

Definition

Definition (Harmonic Bundle)

Four collinear points

A, B, C, D

are in harmonic division (or form a harmonic range) if:

\frac{A C}{CB} = - \frac{A D}{D B}

or equivalently:

\frac{A C}{CB} = \frac{A D}{B D}

(Distance ratio from

C

equals distance ratio from

D

Notation:

(A, B; C, D) = - 1

. We say

C

and

D

are harmonic conjugates with respect to

A

and

B

The Angle Bisector Connection

△ A BX

The internal bisector of $∠ X$ meets $A B$ at $C$ .
The external bisector of $∠ X$ meets extension of $A B$ at $D$ .

Then

(A, B; C, D) = - 1

Apollonius Circle

The locus of points

P

such that

P A / PB = k

(

k \neq = 1

) is a circle. The diameter of this circle is the segment

C D

where

C, D

divide

A B

harmonically in ratio

k

Practice Problems

Exercise (Problem 1)

△ A BC

, let

A D

be the altitude. If

∠ A = 9 0^{\circ}

, prove that

B, C

separate

D

and the foot of the bisector harmonically? (Check exact statement).

Exercise (Problem 2)

Given segment

A B

and its midpoint

M

. Where is the harmonic conjugate of

M

with respect to

A

and

B

? (Point at infinity).

Exercise (Problem 3)

Let

C, D

divide

A B

harmonically. Prove the relation:

\frac{2}{A B} = \frac{1}{A C} + \frac{1}{A D}

(Wait, this is if $A$ is origin? Standard relation is Newton's: $M A^{2} = MC \cdot M D$ where $M$ is midpoint of $A B$ ).