Difference between revisions of "2024 INMO"

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\text {In} triangle ABC with <math>CA=CB</math>, \text{point E lies on the circumcircle of} \text{triangle ABC such that} <math>\angle ECB=90^\circ</math>. \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.}
 
\text {In} triangle ABC with <math>CA=CB</math>, \text{point E lies on the circumcircle of} \text{triangle ABC such that} <math>\angle ECB=90^\circ</math>. \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==
 
==Solution==
[img]https://i.imgur.com/ivcAShL.png[/img]
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[img]file:///Users/lakshyapamecha/Downloads/INMO%20.png[/img]
 
To Prove: Points E, F, P, C are concyclic
 
To Prove: Points E, F, P, C are concyclic
  
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And because <cmath>\angle EFP+\angle ECP=180^\circ</cmath> Points E, F, P, C are concyclic.  
 
And because <cmath>\angle EFP+\angle ECP=180^\circ</cmath> Points E, F, P, C are concyclic.  
 
Hence proven that the centre of the circumcircle of <math>\triangle EGB</math> lies on the circumcircle of <math>\triangle ECF</math>.
 
Hence proven that the centre of the circumcircle of <math>\triangle EGB</math> lies on the circumcircle of <math>\triangle ECF</math>.
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∼Lakshya Pamecha

Revision as of 13:19, 25 April 2024

==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.}

Solution

[img]file:///Users/lakshyapamecha/Downloads/INMO%20.png[/img] To Prove: Points E, F, P, C are concyclic

Observe: \[\angle CAB=\angle CBA=\angle EGA\] \[\angle ECB=\angle CEG=\angle EAB= 90^\circ\] Notice that \[\angle CBA = \angle FGA\] because $CB \parallel EG$ $\Longrightarrow \angle FAG =\angle FGA \Longrightarrow FA= FG$. Here F is the circumcentre of $\triangle EAG$ because $F$ lies on the Perpendicular bisector of AG $\Longrightarrow F$ is the midpoint of $EG \Longrightarrow FP$ is the perpendicular bisector of $EG$. This gives \[\angle EFP =90^\circ\] And because \[\angle EFP+\angle ECP=180^\circ\] Points E, F, P, C are concyclic. Hence proven that the centre of the circumcircle of $\triangle EGB$ lies on the circumcircle of $\triangle ECF$.

∼Lakshya Pamecha