Isotomic conjugate

In geometry, the isotomic conjugate of a point "P" not on a sideline of triangle "ABC" is constructed as follows: Let "A"', "B"', "C"' be the points in which the lines "AP", "BP", "CP" meet the lines "BC", "CA", "AB", respectively. Reflect "A"'"B"'"C"' in the midpoints of sides "BC", "CA", "AB" to obtain points "A", "B", "C", respectively. The lines "AA", "BB", "CC" meet at a point (this can be proved using Ceva's theorem), and this point is called the "isotomic conjugate" of "P".

If trilinears for "P" are "p" : "q" : "r", then trilinears for the isotomic conjugate of "P" are

:"a"⁻²"p"⁻¹ : "b"⁻²"q"⁻¹ : "c"⁻²"r"⁻¹.

The isotomic conjugate of the centroid of triangle "ABC" is the centroid itself.

Isotomic conjugates of lines are circumconics, and conversely, isotomic conjugates of circumconics are lines. (This property holds for isogonal conjugates as well.)

ee also

* Isogonal conjugate

References

* Robert Lachlan, "An Elementary Treatise on Modern Pure Geometry", Macmillan and Co., 1893, page 57.