Brisse Transforms

From ETC (see the notes just before X(1354))

Suppose that P is a point on the circumcircle Γ of triangle ABC. Let U and V be the lines through P tangent to the incircle. Line U meets Γ in a point U' other than P, and line V meets Γ in a point V' other than P. The line U'V' is tangent to the incircle. The touchpoint, denoted by T(P), is the Brisse transform of P. Suppose P is given by barycentrics u : v : w. Barycentrics for T(P) are found in

Edward Brisse, Perspective Poristic Triangles: a4/[(b + c - a)u2] : b4/[(c + a - b)v2] : c4/[(a + b - c)w2].

If X is given by trilinears x : y : z, then T(X) has trilinears a/[(b + c - a)x2] : b/[(c + a - b)y2] : c/[(a + b - c)z2].

Examples: X(11) = Feuerbach point = T(X(109))
X(1317) = incircle-antipode of X(11) = T(X(106))

Angel Montesdeoca. Diciembre, 2017