Isotomic conjugate

In geometry, the isotomic conjugate of a point Template:Mvar with respect to a triangle △ABC is another point, defined in a specific way from Template:Mvar and △ABC: If the base points of the lines Template:Mvar on the sides opposite Template:Mvar are reflected about the midpoints of their respective sides, the resulting lines intersect at the isotomic conjugate of Template:Mvar.

Construction

File:Isotomic conjugate of a point.svg

We assume that Template:Mvar is not collinear with any two vertices of △ABC. Let Template:Mvar be the points in which the lines Template:Mvar meet sidelines Template:Mvar (extended if necessary). Reflecting Template:Mvar in the midpoints of sides Template:Mvar will give points Template:Mvar respectively. The isotomic lines Template:Mvar joining these new points to the vertices meet at a point (which can be proved using Ceva's theorem), the isotomic conjugate of Template:Mvar.

If the trilinears for Template:Mvar are Template:Mvar, then the trilinears for the isotomic conjugate of Template:Mvar are

$a^{-2}{p}^{-1}:b^{-2}{q}^{-1}:c^{-2}{r}^{-1},$

where Template:Mvar are the side lengths opposite vertices Template:Mvar respectively.

Properties

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

The isotomic conjugate of the symmedian point is the third Brocard point, and the isotomic conjugate of the Gergonne point (whose Cevian triangle is the intouch triangle) is the Nagel point (whose Cevian triangle is the extouch triangle).

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

See also

• Isogonal conjugate
• Triangle center

References

• Robert Lachlan, An Elementary Treatise on Modern Pure Geometry, Macmillan and Co., 1893, page 57.
• Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007, Template:ISBN, pp. 157–159, 278

External links

• Pauk Yiu: Isotomic and isogonal conjugates
• Navneel Singhal: Isotomic and isogonal conjugates