WOLFRAM|DEMONSTRATIONS PROJECT

The Fourth Harmonic Point of a Triangle

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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.