### 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**: a^{4}/[(b + c - a)u^{2}] : b^{4}/[(c
+ a - b)v^{2}] : c^{4}/[(a + b - c)w^{2}].

If X is given by trilinears x : y : z, then T(X) has trilinears a/[(b + c - a)x^{2}] :
b/[(c + a - b)y^{2}] : c/[(a + b - c)z^{2}].

Examples: X(11) = Feuerbach point = T(X(109))

X(1317) = incircle-antipode of X(11) = T(X(106))

Angel Montesdeoca. Diciembre, 2017