The Fourth Harmonic Point of a Triangle

angle θ

The fourth harmonic point of a triangle is an invariant point under a certain geometric transformation. Given a triangle

PAB

with

A

and

B

fixed, extend the segment

BA

to another fixed point

O

. Draw a line through

O

to intersect the line

PA

at

D

and the line

PB

at

C

. Let

F

be the intersection of the lines

BD

and

CA

. Let the line

PF

intersect

AB

at

E

. Then

E

, called the fourth harmonic point, is invariant either by moving

P

or changing the slope of the line

ODC

.

The proof follows from Ceva's theorem and Menelaus's theorem, which shows that the ratio of the length of

AE

to that of

EB

is constant and equals

BA/BO

. Simple as it is, the example also reveals the duality of

P

and

F

, which is the one of the most important concepts in projective geometry.