The Fourth Harmonic Point of a Triangle
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 .[more]
The proof follows from Ceva's theorem and Menelaus's theorem, which shows that the ratio of the length of to that of is constant and equals . Simple as it is, the example also reveals the duality of and , which is the one of the most important concepts in projective geometry.[less]
 J. Gray, Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century, London: Springer-Verlag, 2007.
 L. Cremona, Elements of Projective Geometry, 3rd ed. (C. Leudesdorf, trans.), New York: Dover, 1960.
 R. A. Johnson, Modern Geometry, New York: Houghton–Mifflin, 1929.