Problem 4! IMO 2015

Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$ . A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$ , such that $B$ , $D$ , $E$ , and $C$ are all different and lie on line $B$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$ , such that $A$ , $F$ , $B$ , $C$ , and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$ . Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $AC$ .

Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$ . Prove that $X$ lies on the line $AO$ .

This is part of the set IMO 2015

#Geometry #Circumcircle #InternationalMathOlympiad(IMO) #Circles #IMO2015

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

This is a problem which looks initially scary with all the circles, but is just a chasing down of angles. Work backwards from the result, and see what we need. E.g. If X lies on AO, what does that tell us about X?

Calvin Lin Staff - 5 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$