Pedal Triangles of Isogonal Conjugates

Let ABC be a triangle and P be a point. The reflections of the three lines AP, BP, and CP in the angle bisectors at A, B, and C meet in a point I, called the isogonal conjugate of P.
The feet of the perpendiculars from P to the sides of triangle ABC form the pedal triangle of P, RST. Similarly, let the pedal triangle of I be UVW.
Then R, S, T, U, V, W all lie on a circle.

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The theorem is stated in:
N. A. Court, "Isogonal Conjugate Points for a Triangle," The Mathematical Gazette, 36(317), 1952 pp. 167–170.
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