Isotomic conjugate

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In geometry, the isotomic conjugate of a point P not on a sideline of triangle ABC is constructed as follows: Let A', B', C' be the points in which the lines AP, BP, CP meet the lines BC, CA, AB, respectively. Reflect A'B'C' in the midpoints of sides BC, CA, AB to obtain points A", B", C", respectively. The lines AA", BB", CC" meet at a point (this can be proved using Ceva's theorem), and this point is called the isotomic conjugate of P.

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

a⁻²p⁻¹ : b⁻²q⁻¹ : c⁻²r⁻¹.

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

Isotomic conjugates of lines are circumconics, and conversely, isotomic conjugates of circumconics are lines. (This property holds for isogonal conjugates as well.)

[edit] See also

• Isogonal conjugate

[edit] References

• Robert Lachlan, An Elementary Treatise on Modern Pure Geometry, Macmillan and Co., 1893, page 57.