10217

# 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 .
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.

### DETAILS

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.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.