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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 (
''Isotomic and isogonal conjugates''
* Navneel Singhal
''Isotomic and isogonal conjugates''
Triangle geometry zh:等角共轭
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, 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 (extended
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* Exte ...
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 kn ...
), 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 the centroid of triangle is the centroid itself. The isotomic conjugate of the symmedian point is the thirdBrocard 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 ''ABC'' with sides ''a'', ''b'', and ''c'', where the vertices are labeled '' ...
, 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. (This property holds for isogonal conjugates as well.)
See also
* Isogonal conjugate * Triangle centerReferences
* 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