Isotomic conjugate
In geometry, the isotomic conjugate of a point with respect to a triangle is another point, defined in a specific way from and : If the base points of the lines on the sides opposite are reflected about the midpoints of their respective sides, the resulting lines intersect at the isotomic conjugate of.
Construction
We assume that is not collinear with any two vertices of. Let be the points in which the lines meet sidelines . Reflecting in the midpoints of sides will give points respectively. The isotomic lines joining these new points to the vertices meet at a point, the isotomic conjugate of.Coordinates
If the trilinears for are, then the trilinears for the isotomic conjugate of arewhere are the side lengths opposite vertices respectively.
Properties
The isotomic conjugate of the centroid of triangle is the centroid itself.The isotomic conjugate of the symmedian point is the third Brocard point, and the isotomic conjugate of the Gergonne point is the Nagel point.
Isotomic conjugates of lines are circumconics, and conversely, isotomic conjugates of circumconics are lines.