## Isogonal Conjugation

Let t = (ABC) be a triangle. For each point D on its plane consider the symmetrics of DA, AB, AC with respect to the corresponding bisectors. The resulting lines EA, EB, EC concur at a point E, called the Isogonal-Conjugate of D.
The map f: D ---> E is the isogonal conjugation, defined by the triangle t.
f is involutive i.e. f2 = 1.

[1] The concurrence of lines at E can be proved easily using Ceva's theorem. For the trace D1 on BC it is
D1B/D1C=(DJ/sin(B))/(DH/sin(C)) = k1(sin(C)/sin(B)).
The corresponding trace E1 on BC has
E1B/E1C=(EQ/sin(B))/(EO/sin(C))=(1/k1)(sin(C)/sin(B)).
Later because of the symmetry on the bisector. By Ceva the product of the analogous ratios for all sides k1k2k3=1 implies that the corresponding ratios relative to E have also (1/k1)(1/k2)(1/k3)=1.
[2] The six projections of D and its isogonal E on the sides lie on a circle of radius r (the circumcircle of the triangle with vertices I, J, H, the projections of D on the sides of t). This is seen easily by considering the middle U of DE and identifying it with the center of the circle.
[3] Triangle t(D) = IJH is called the pedal triangle of point D with respect to triangle t. Its circumcircle c(U,r) is called the pedal circle of point D. Notice that triangles t(D), t(E) share the same pedal circle c(U,r).
[4] The symmetry of D, E on U implies that P*, diametral of P, has DP* parallel and equal to PE.
[5] This implies that |DI||EP| = r² - |DU|² = k (the power of D with respect to the circle), thus leading to the relation of the trilinear coordinates of E: (x*, y*, z*) = ( k/x, k/y, k/z ) to those of D.

The isogonal conjugation, given in trilinears by f : (x,y,z) -->(k/x, k/y, k/z), is not properly defined on the vertices of the triangle. Outside the sides (and their prolongations) f is one-to-one (invertible). f sends all the points of a side of the triangle to the opposite vertex.
The isogonal conjugation is a special case of the more general quadratic transforms related to families of conics (see IsogonalAsQuadratic.html ).
There are several applications of this transform. A non standard one can be found in the file: Isogonal_3TangentCircles.html .

Some remarkable triangle centers and their isogonal conjugates, {a,b,c} denoting the side-lengths of the triangle and {A,B,C} the angle-measures:

X1 : Incenter: (1,1,1) ----> Incenter, fixed point of isogonal conjugation.
X2 : Centroid: (1/a, 1/b, 1/c) ----> X6 : Symmedian point (a, b, c).
X3 : Circumcenter (cosA, cosB, cosC) ----> X4 : Orthocenter (1/cosA, 1/cosB, 1/cosC).
B1 : First Brocard point (c/b, a/c, b/a) ----> B2 : Second Brocard point (b/c, c/a, a/b).
X7 : Gergonne point (b*c/(b+c-a), ... ) ----> X55 : Internal center of similitude circum-/ in-circle (a*(b+c-a), ...).
X8 : Nagel point ((b+c-a)/a, ...) ----> X56 : External center of similitude circum-/ in-circle (a/(b+c-a),...).
The periods in the definitions of coordinates mean a cyclic permutation of {a,b,c}.