Difference between revisions of "2024 INMO"
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Observe: <cmath>\angle CAB=\angle CBA=\angle EGA</cmath> <cmath>\angle ECB=\angle CEG=\angle EAB= 90^\circ</cmath> | Observe: <cmath>\angle CAB=\angle CBA=\angle EGA</cmath> <cmath>\angle ECB=\angle CEG=\angle EAB= 90^\circ</cmath> | ||
− | Notice that <cmath>\angle CBA = \angle FGA</cmath> because <math>CB \parallel EG</math> <math>\Longrightarrow \angle FAG =\angle FGA | + | Notice that <cmath>\angle CBA = \angle FGA</cmath> because <math>CB \parallel EG</math> <math>\Longrightarrow \angle FAG =\angle FGA \Longrightarrow FA= FG</math>. |
− | + | Here F is the circumcentre of <math>\triangle EAG</math> because <math>F</math> lies on the Perpendicular bisector of AG | |
− | + | <math>\Longrightarrow F</math> is the midpoint of <math>EG \Longrightarrow FP</math> is the perpendicular bisector of <math>EG</math>. | |
− | + | This gives <cmath>\angle EFP =90^\circ</cmath> | |
− | This gives<cmath>\angle EFP =90^\circ</cmath> | + | 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> | ||
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>. |
Revision as of 14:12, 25 April 2024
==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
.
Here F is the circumcentre of
because
lies on the Perpendicular bisector of AG
is the midpoint of
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
.