geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, 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 Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. The ...

about the

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dim ...

s 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 (

extended Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (proof theory) * Extension (predicate logic), the set of tuples of values that ...

if necessary). Reflecting in the midpoints of sides will give points respectively. The isotomic lines joining these new points to the vertices meet at a point (which can be proved using

Ceva's theorem In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle , let the lines be drawn from the vertices to a common point (not on one of the sides of ), to meet opposite sides at respectively. (The segments are k ...

), the ''isotomic conjugate'' of .

Coordinates

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

a^p^ : b^q^ : c^r^,

where are the side lengths opposite vertices respectively.

Properties

The isotomic conjugate of the

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...

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

symmedian point In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the co ...

is the third

Brocard point In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician. Definition In a triangle with sides , where the vertices are labeled in counterclockwise order, ther ...

, and the isotomic conjugate of the

Gergonne point In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

(whose Cevian triangle is the intouch triangle) is the

Nagel point In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concur ...

(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.)

References

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

External links

*{{mathworld, id=IsotomicConjugate, title=Isotomic Conjugate * Pauk Yiu
''Isotomic and isogonal conjugates''
* Navneel Singhal
''Isotomic and isogonal conjugates''
Triangle geometry zh:等角共轭

Construction

Coordinates

Properties

See also

References

External links