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

Construction

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We assume that P is not collinear with any two vertices of ABC. Let A', B', C' be the points in which the lines AP, BP, CP meet sidelines BC, CA, AB (extended if necessary). Reflecting A', B', C' in the midpoints of sides BC, CA, AB will give points A", B", C" respectively. The isotomic lines AA", BB", CC" joining these new points to the vertices meet at a point (which can be proved using Ceva's theorem), the isotomic conjugate of P.

Coordinates

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If the trilinears for P are p : q : r, then the trilinears for the isotomic conjugate of P are

 

where a, b, c are the side lengths opposite vertices A, B, C respectively.

Properties

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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 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.)

Generalization

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X-Dao conjugate of P

In may 2021, Dao Thanh Oai given a generalization of Isotomic conjugate as follows:[1]

Let △ ABC be a triangle, P be a point on its plane and Ω an arbitrary circumconic of △ ABC. Lines AP, BP, CP cuts again Ω at A', B', C' respectively, and parallel lines through these points to BC, CA, AB cut Ω again at A", B", C" respectively. Then lines AA", BB", CC" concurent.

If barycentric coordinates of the center X of Ω are   and  , then D, the point of intersection of AA", BB", CC" is:  

The point D above call the X-Dao conjugate of P, this conjugate is a generalization of all known kinds of conjugaties:[1]

  • When Ω is the circumcircle of ABC, Dao conjugate become the isogonal conjugate of P.
  • When Ω is the Steiner circumellipse of ABC, Dao conjugate become the isotomic conjugate of P.
  • When Ω is the circumconic of ABC with center X = X(1249), Dao conjugate become the Polar conjugate of P.

See also

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References

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  • Robert Lachlan, An Elementary Treatise on Modern Pure Geometry, Macmillan and Co., 1893, page 57.
  • Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007, ISBN 978-0-486-46237-0, pp. 157–159, 278
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