Question

Let ABC be an acute-angled triangle with AB < AC. Let I be the incentre of triangle ABC, and let D, E, F be the points at which its incircle touches the sides BC, CA, AB, respectively. Let BI, CI meet the line EF at Y, X, respectively. Further assume that both X and Y are outside the triangle ABC. Prove that 

(i) B, C, Y, X are concyclic; and 

(ii) I is also the incentre of triangle DYX.

Answer 1

Construction : Join XB, Join XD, Join DY. Let IY intersect AL at k. Let AI cuts EF at L