2024 INMO

==Problem 1

\text {In} triangle ABC with $CA=CB$ , \text{point E lies on the circumcircle of} \text{triangle ABC such that} $\angle ECB=90^\circ$ . \text{The line through E parallel to CB intersect CA in F} \text{and AB in G}.\text{Prove that}\\ \text{the centre of the circumcircle of} triangle EGB \text{lies on the circumcircle of triangle ECF.}