2024 INMO
==Problem 1
\text {In} triangle ABC with , \text{point E lies on the circumcircle of} \text{triangle ABC such that} . \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.}
Solution
To Prove: Points E,F,P,C are concyclic \newpage
Observe: Notice that because
or .
\Here F is the circumcentre of \traingle EAG becuase F lies on the Perpendicular bisector of AG.\\\\ \implies is the midpoint of \implies is the perpendicular bisector of .\\ This gives.\\ And because . Points E,F,P,C are concyclic.\\ Hence proven that the centre of the circumcircle of lies on the circumcircle of .