The Fourth Harmonic Point of a Triangle

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The fourth harmonic point of a triangle is an invariant point under a certain geometric transformation. Given a triangle with
and
fixed, extend the segment
to another fixed point
. Draw a line through
to intersect the line
at
and the line
at
. Let
be the intersection of the lines
and
. Let the line
intersect
at
. Then
, called the fourth harmonic point, is invariant either by moving
or changing the slope of the line
.
Contributed by: Shenghui Yang (June 2012)
Open content licensed under CC BY-NC-SA
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References
[1] J. Gray, Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century, London: Springer-Verlag, 2007.
[2] L. Cremona, Elements of Projective Geometry, 3rd ed. (C. Leudesdorf, trans.), New York: Dover, 1960.
[3] R. A. Johnson, Modern Geometry, New York: Houghton–Mifflin, 1929.
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