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 1
\includegraphics[width=1.25\textwidth]{INMO 2024 P1.png} To Prove: Points E,F,P,C are concyclic \newpage
Observe:
\text{Notice that}
because
} \implies
\:\text{or} \:.\\
\text{Here F is the circumcentre of \traingle EAG becuase F lies on the Perpendicular bisector of AG.}\\\\
\implies \text{ is the midpoint of
} \implies \text{
is the perpendicular bisector of
.}\\
\text{This gives} \:
.\\
\text{And because}
. \:\text{Points E,F,P,C are concyclic.}\\
\text{Hence proven that the centre of the circumcircle of
lies on the circumcircle of
.}