Construction
We assume that P is not collinear with any two vertices of ABC. Let A', B' and C' be the points in which the lines AP, BP, CP meet sidelines BC, CA and AB (extended if necessary). Reflecting A', B', C' in the midpoints of sides BC, CA, AB will give points A", B" and C" respectively. The isotomic lines AA", BB" and 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.
Read more about this topic: Isotomic Conjugate
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