2016 OIM Problems/Problem 3

Problem

Let $ABC$ be an acute triangle whose circumcircle is $\Gamma$. The tangents to $\Gamma$ through $B$ and $C$ intersect at $P$. On the arc $AC$ that does not contain $B$, we get a point $M$, different from $A$ and $C$, such that the line $AM$ cuts the line $BC$ at $K$. Let $R$ be the symmetric point of $P$ with respect to the line $AM$, and $Q$ the point of intersection of the lines $RA$ and $PM$. Let $J$ be the midpoint of $BC$ and $L$ be the point where the line parallel by $A$ to the line $PR$ intersects the line $PJ$. Show that the points $L, J, A, Q$ and $K$ are on the same circle.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

OIM Problems and Solutions