IsogonalAsQuadratic

Isogonal transformation as a quadratic one

Quadratic transformations are related to families of conics. Given a family of conics, and a point P, all polars of P with respect to the conic-members of the family pass through another point P' = F(P). This defines the quadratic transformation of the family. In practice, P' is constructed as the intersection point of two polars L₁, L₂ of P with respect to two fixed members of the family.

A particular case of quadratic transformation is the isogonal conjugation Q' = I(Q) with respect to a triangle ABC. This is defined geometrically by taking the cevians of Q and their symmetrics with respect to the bisectors of the triangle. Using Ceva's theorem one shows that these symmetrics pass through a point Q' which is defined as the isogonal conjugate of Q.

Here we show that the isogonal transformation of ABC coincides with the quadratic transformation of a special family of conics consisting entirely of rectangular hyperbolas. The family namely consists of all conics passing through the four intersection points {I,A',B',C'} of the bisectors of the triangle. All these conics are rectangular hyperbolas. Their family is generated by the linear combinations of two degenerate members c₁, c₂, consisting of the two bisectors at A (degenerate conic c₁(AA',AB')) and the two bisectors at B (degenerate conic c₂(BA',BB')).

The proof, modulo the remarks made above, is trivial. Apply the definition of the quadratic transformation by taking the polars of P with respect to c₁, c₂. For example, the polar L₁ = AP' of P with respect to c₁(AA',AB') and line L = AP are harmonic conjugates with respect to AA', AB', which are orthogonal. Hence (see Harmonic_Bundle.html ) the angle of (L, L₁) is bisected by lines (AA', AB'). Analogous results are obtained by joining P, P' with the remaining vertices B and C. This completes the proof.
Although the property has a simple proof. The context involves several interesting ideas studied in the references given below.

Isogonal transformation as a quadratic one

See Also