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We assume that is not collinear with any two vertices of . Let be the points in which the lines meet sidelines ( extended 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), the ''isotomic conjugate'' of .
''Isotomic and isogonal conjugates''
* Navneel Singhal
''Isotomic and isogonal conjugates''
Triangle geometry zh:等角共轭
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 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), the ''isotomic conjugate'' of .
Coordinates
If the trilinears for are , then the trilinears for the isotomic conjugate of are : where are the side lengths opposite vertices respectively.Properties
The isotomic conjugate of thecentroid
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 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
In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, ...
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, 278External 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