Isogonal Conjugates

The isogonal conjugate of a point P with respect to triangle ABC is constructed by reflecting each of the three cevians (lines from vertex to opposite side through P) across the corresponding angle bisector. The three reflected lines meet at a single point P′ — this is a theorem, not a coincidence. The algebraic reason is that reflection across the angle bisector swaps the two sides of an angle, and the condition for three cevians to be concurrent (Ceva's theorem) is preserved under this operation.

The most celebrated isogonal pair is the orthocenter H and the circumcenter O. When P equals the orthocenter (intersection of altitudes), its isogonal conjugate is precisely the circumcenter (center of the circumscribed circle). This is a profound duality: the altitude from A is the reflection of the line AO across the angle bisector at A. Another famous pair is the centroid G and the symmedian point K (also called the Lemoine point). The symmedian through vertex A is the reflection of the median across the angle bisector — it passes through the midpoint of BC in the “most symmetric” sense with respect to angles.

The incenter I is its own isogonal conjugate — it lies on all three angle bisectors, so reflecting a cevian through I across a bisector returns the same line. This makes the incenter the unique fixed point of the isogonal conjugation map. Points on the circumcircle have conjugates “at infinity” (the reflected cevians become parallel), which is why the conjugate map is only defined in the interior of the triangle in the classical sense.

Isogonal conjugates appear throughout olympiad geometry because the transformation is an involution (applying it twice returns to P) and preserves many metric and projective properties. The conjugate of a circle through the vertices is another circle; the conjugate of a line is a circumconic. These facts make isogonal conjugation one of the most powerful tools in triangle geometry, appearing in problems from the IMO to Putnam competitions.